Try to Solve This Problem … without Algebra

Can you solve the following, without doing any Algebraic manipulation?  Just by reading and thinking about what it says, can you figure out what x is?  (The numbers a, x, andare not zero.)

Given:  3ak

And:  ax = 4k

What must x be?

If you’re versed much at all in basic Algebra you will be tempted to substitute and solve.  After all, this is a system of equations.  But that will bypass the purpose and benefit of the exercise.

The intended benefit of this problem is that it promotes mathematical literacy, in particular, seeing relationships between terms.  It’s not a complicated relationship but it is of utmost importance to this problem.  Once you read and make sense of what the mathematical relationships are you can talk your way through the problem.

Once again, I believe the purpose of homework is learning.  Sure, sometimes it is practice and familiarity, but those are the only times that answer-getting is important.  Without understanding, having the right answer is often of little to no use.  If it were, then copying the answers from the back of the book would be sufficient for learning, right?

If you’re ready to see the solution, you can watch the video or read the text after the video.

 

I understand that sometimes it’s appropriate to read, but not listen or watch a video.  So here’s how this works.

Given:  3ak

And:  ax = 4k

The first statement says that the number k is three times bigger than the number a.  We don’t know what or k are but we know how they’re related and can think of lots of numbers that fit this relationship.  One number that’s three times as large as the other.

The number k is three times as big as the number a.

Think of this relationship one more way, for a moment.  The number k has two factors, 3 and a.  Whether is composite or prime is irrelevant really, it won’t change the fact that we could write k as the product of two numbers.  I mention this, not because it helps solve this problem but because it might.  Without knowing the path, sometimes it is a good idea to brain storm for a  moment and list as many things you know about the information given, before seeking an answer.  Sometimes, doing so, makes the answer apparent to you!

Let’s look at the second statement now.

Another number times a is four times as big as k.  This is perhaps a bit distracting, but the key information is there.  Remember, k is three times as big as a.  Now we have something four times larger than k.

Let’s look at this a different way.  The number 4k is not k at all, but instead, k and 4 are factors of new number.

If this new number is four times larger than k, and k is three times larger than a, how much larger is this new number than a?

You have three times as much money as me.  Bobert has four times as much as you do.  How much more money does Bob have than me?

For every dollar I have you have three.  For every dollar you have, Bobert has four.

Still don’t see it?  I know…picture good, word bad.  Here you go.

You have three times as much as I do. For every one dollar I have, you have three.
For every dollar you have, Bobert has four.

If 3ak, and ax = 4k, then is 12 because

Exponents Part 2

two

Exponents Part 2

Division

In the previous section we learned that exponents are repeated multiplication, which on its own is not tricky. What makes exponents tricky is determining what is a base and what is not for a given exponent. It is imperative that you really understand the material from the previous section before tackling what’s next. If you did not attempt the practice problems, you need to. Also watch the video that review them.

In this section we are going to see why anything to the power of zero is one and how to handle negative exponents, and why they mean division.

What Happens with Division and Exponents?

Consider the following expression, keeping in mind that the base is arbitrary, could be any number (except zero, which will be explained soon).

3 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGynaaaaaaa@37A0@

This equals three times itself five total times:

3 5 =33333 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGynaaaakiabg2da9iaaiodacqGHflY1caaIZaGaeyyX ICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaaaa@4589@

Now let’s divide this by 3. Note that 3 is just 31.

3 5 3 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGymaaaaaaaaaa@395E@

If we write this out to seek a pattern that we can use for a short-cut, we see the following:

3 5 3 1 = 33333 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGymaaaaaaGccqGH9aqpdaWcaaqaaiaaiodacqGHflY1caaIZaGaey yXICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaabaGaaG4maaaa aaa@4814@

If you recall how we explored reducing Algebraic Fractions, the order of division and multiplication can be rearranged, provided the division is written as multiplication of the reciprocal. That is how division is written here.

3 5 3 1 = 3 3 3333 1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGymaaaaaaGccqGH9aqpdaWcaaqaaiaaiodaaeaacaaIZaaaaiabgw SixpaalaaabaGaaG4maiabgwSixlaaiodacqGHflY1caaIZaGaeyyX ICTaaG4maaqaaiaaigdaaaaaaa@48DF@

And of course 3/3 is 1, so this reduces to:

3 5 3 1 =3333= 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGymaaaaaaGccqGH9aqpcaaIZaGaeyyXICTaaG4maiabgwSixlaaio dacqGHflY1caaIZaGaeyypa0JaaG4mamaaCaaaleqabaGaaGinaaaa aaa@46EE@

The short-cut is:

3 5 3 1 = 3 51 = 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGymaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aGaeyOeI0 IaaGymaaaakiabg2da9iaaiodadaahaaWcbeqaaiaaisdaaaaaaa@4077@

That is, if the bases are the same you can reduce. Reducing eliminates one of the bases that is being multiplied by itself from both the numerator and the denominator. A general form of the third short-cut is here:

Short-Cut 3: a m a n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbWaaWbaaSqabeaacaWGTbaaaaGcbaGaamyyamaaCaaaleqabaGa amOBaaaaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaeyOeI0 IaamOBaaaaaaa@3F10@

This might seem like a worthless observation, but this will help articulate the very issue that is going to cause trouble with exponents and division.

3 5 3 1 = 3 5 ÷ 3 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGymaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aaaaOGaey 49aGRaaG4mamaaCaaaleqabaGaaGymaaaaaaa@4002@ .

But that is different than

3 1 ÷ 3 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGymaaaakiabgEpa4kaaiodadaahaaWcbeqaaiaaiwda aaaaaa@3B8A@

The expression above is the same as

3 1 3 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaIXaaaaaGcbaGaaG4mamaaCaaaleqabaGa aGynaaaaaaaaaa@395F@

This comes into play because

3 1 3 5 = 3 15 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaIXaaaaaGcbaGaaG4mamaaCaaaleqabaGa aGynaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaIXaGaeyOeI0 IaaGynaaaaaaa@3DC0@ ,

and 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=nbiaaa@37C3@ 5 = -4.

Negative Exponents?

In one sense, negative means opposite. Exponents mean multiplication, so a negative exponent is repeated division. This is absolutely true, but sometimes difficult to write out. Division is not as easy to write as multiplication.

Consider that 3-4 is 1 divided by 3, four times. 1 ÷ 3 ÷ 3 ÷ 3 ÷ 3. But if we rewrite each of those ÷ 3 as multiplication by the reciprocal (1/3), it’s must cleaner and what happens with a negative exponent is easier to see.

1÷3÷3÷3÷31 1 3 1 3 1 3 1 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgE pa4kaaiodacqGH3daUcaaIZaGaey49aGRaaG4maiabgEpa4kaaioda cqGHsgIRcaaIXaGaeyyXIC9aaSaaaeaacaaIXaaabaGaaG4maaaacq GHflY1daWcaaqaaiaaigdaaeaacaaIZaaaaiabgwSixpaalaaabaGa aGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaIXaaabaGaaG4maa aaaaa@5482@

This is classically repeated multiplication. While one times itself any number of times is still one, let’s go ahead and write it out this time.

1 1 3 1 3 1 3 1 3 1 ( 1 3 ) 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgw SixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaI XaaabaGaaG4maaaacqGHflY1daWcaaqaaiaaigdaaeaacaaIZaaaai abgwSixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyOKH4QaaGymaiab gwSixpaabmaabaWaaSaaaeaacaaIXaaabaGaaG4maaaaaiaawIcaca GLPaaadaahaaWcbeqaaiaaisdaaaaaaa@4EE8@

This could also be written:

1 1 3 1 3 1 3 1 3 1 1 4 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgw SixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyyXIC9aaSaaaeaacaaI XaaabaGaaG4maaaacqGHflY1daWcaaqaaiaaigdaaeaacaaIZaaaai abgwSixpaalaaabaGaaGymaaqaaiaaiodaaaGaeyOKH4QaaGymaiab gwSixpaalaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaOqaaiaaio dadaahaaWcbeqaaiaaisdaaaaaaaaa@4E54@

The second expression is easier, but both are shown here to make sure you see they are the same.

Since 1 times 14 is just one, we can simplify this further to:

1 1 4 3 4 = 1 3 4 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiabgw SixpaalaaabaGaaGymamaaCaaaleqabaGaaGinaaaaaOqaaiaaioda daahaaWcbeqaaiaaisdaaaaaaOGaeyypa0ZaaSaaaeaacaaIXaaaba GaaG4mamaaCaaaleqabaGaaGinaaaaaaGccaGGUaaaaa@40A3@

Negative exponents are repeated division. Since division is hard to write and manipulate, we will write negative exponents as multiplication of the reciprocal. In fact, if instructions say to simplify, you cannot have a negative exponent in your final answer. You must rewrite it as multiplication of the reciprocal. Sometimes that can get ugly. Consider the following:

b a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@39AD@

To keep this clean, let us consider separating this single fraction as the product of two rational expressions.

b a 5 = b 1 1 a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH 9aqpdaWcaaqaaiaadkgaaeaacaaIXaaaaiabgwSixpaalaaabaGaaG ymaaqaaiaadggadaahaaWcbeqaaiabgkHiTiaaiwdaaaaaaaaa@4243@

The b is not a problem here, but the other rational expression is problematic. We need to multiply by the reciprocal of 1 a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIXaaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@3981@ , which is just a5.

b a 5 = b 1 a 5 1 = a 5 b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGIbaabaGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH 9aqpdaWcaaqaaiaadkgaaeaacaaIXaaaaiabgwSixpaalaaabaGaam yyamaaCaaaleqabaGaaGynaaaaaOqaaiaaigdaaaGaeyypa0Jaamyy amaaCaaaleqabaGaaGynaaaakiaadkgaaaa@4529@ .

This can also be considered a complex fraction, the likes of which we will see very soon. Let’s see how that works.

b a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGIbaaqqaaaaaaOpGqSvxza8qabaGaamyyamaaCaaaleqabaGaeyOe I0IaaGynaaaaaaaaaa@3C44@

Note: a 5 = 1 a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeaaaaaa6di eB1vgapeGaamyyamaaCaaaleqabaGaeyOeI0IaaGynaaaak8aacqGH 9aqpqqa6daaaaaGuLrgapiWaaSaaaeaacaaIXaaabaGaamyyamaaCa aaleqabaGaaGynaaaaaaaaaa@4134@

Substituting this we get:

b 1 a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGIbaaqqa6daaaaaGuLrgapeqaamaalaaabaGaaGymaaqaaiaadgga daahaaWcbeqaaiaaiwdaaaaaaaaaaaa@3BB5@

This is b divided by 1/a5.

b÷ 1 a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgE pa4oaalaaabaGaaGymaaqaaiaadggadaahaaWcbeqaaiaaiwdaaaaa aaaa@3BB6@

Let’s multiply by the reciprocal:

b a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOyaiabgw SixlaadggadaahaaWcbeqaaiaaiwdaaaaaaa@3AFA@

Now we will rewrite it in alphabetical order (a good habit, for sure).

a 5 b MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGynaaaakiaadkgaaaa@38BA@

Let us consider one more example before we make our fourth short-cut. With this example we could actually apply our second short-cut, but it will not offer much insight into how these exponents work with division.

This is the trickiest of all of the ways in which exponents are manipulated, so it is worth the extra exploration.

2 x 2 y 5 z 2 2 x y 3 z 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaamiEamaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaadMhadaah aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale qabaGaeyOeI0IaaGOmaaaakiaadIhacaWG5bWaaWbaaSqabeaacaaI ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaaaaa@45E3@

As you see we have four separate bases. In order to simplify this expression we need one of each base (2, x, y, z), and all positive exponents. So let’s separate this into the product of four rational expressions, then simplify each.

2 x 2 y 5 z 2 2 x y 3 z 5 2 2 2 x 2 x y 5 y 3 z z 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaamiEamaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaadMhadaah aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale qabaGaeyOeI0IaaGOmaaaakiaadIhacaWG5bWaaWbaaSqabeaacaaI ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGHsg IRdaWcaaqaaiaaikdaaeaacaaIYaWaaWbaaSqabeaacqGHsislcaaI YaaaaaaakiabgwSixpaalaaabaGaamiEamaaCaaaleqabaGaeyOeI0 IaaGOmaaaaaOqaaiaadIhaaaGaeyyXIC9aaSaaaeaacaWG5bWaaWba aSqabeaacqGHsislcaaI1aaaaaGcbaGaamyEamaaCaaaleqabaGaaG 4maaaaaaGccqGHflY1daWcaaqaaiaadQhaaeaacaWG6bWaaWbaaSqa beaacqGHsislcaaI1aaaaaaaaaa@5ED4@

The base of two first:

2 2 2 2÷ 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaaabaGaaGOmamaaCaaaleqabaGaeyOeI0IaaGOmaaaaaaGccqGH sgIRcaaIYaGaey49aGRaaGOmamaaCaaaleqabaGaeyOeI0IaaGOmaa aaaaa@40D5@

We wrote it as division. What we will see is dividing is multiplication by the reciprocal, and then the negative exponent is also dividing, which is multiplication by the reciprocal. The reciprocal of the reciprocal is just the original. But watch what happens with the sign of the exponent.

First we will rewrite the negative exponent as repeated division.

2÷ 1 2 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgE pa4oaalaaabaGaaGymaaqaaiaaikdadaahaaWcbeqaaiaaikdaaaaa aaaa@3B5E@

Now we will rewrite division as multiplication by the reciprocal.

2 2 2 = 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiabgw SixlaaikdadaahaaWcbeqaaiaaikdaaaGccqGH9aqpcaaIYaWaaWba aSqabeaacaaIZaaaaaaa@3D58@

Keep in mind, this is the same as 23/1.

We will offer similar treatment to the other bases.

Consider first x 2 x = x 2 1 1 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaamiEaaaacqGH 9aqpdaWcaaqaaiaadIhadaahaaWcbeqaaiabgkHiTiaaikdaaaaake aacaaIXaaaaiabgwSixpaalaaabaGaaGymaaqaaiaadIhaaaaaaa@42A1@

Negative exponents are division, so:

x 2 x = x 2 1 1 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaamiEaaaacqGH 9aqpdaWcaaqaaiaadIhadaahaaWcbeqaaiabgkHiTiaaikdaaaaake aacaaIXaaaaiabgwSixpaalaaabaGaaGymaaqaaiaadIhaaaaaaa@42A1@

Notice the x that is already dividing (in the denominator) does not change. It has a positive exponent, which means it is already written as division.

x 2 1 1 x 1 x 2 1 x = 1 x 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WG4bWaaWbaaSqabeaacqGHsislcaaIYaaaaaGcbaGaaGymaaaacqGH flY1daWcaaqaaiaaigdaaeaacaWG4baaaiabgkziUoaalaaabaGaaG ymaaqaaiaadIhadaahaaWcbeqaaiaaikdaaaaaaOGaeyyXIC9aaSaa aeaacaaIXaaabaGaamiEaaaacqGH9aqpdaWcaaqaaiaaigdaaeaaca WG4bWaaWbaaSqabeaacaaIZaaaaaaaaaa@4A23@

This is exactly how simplifying the y and z will operation.

2 3 1 1 x 2 x 1 y 5 y 3 z z 5 1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaWaaWbaaSqabeaacaaIZaaaaaGcbaGaaGymaaaacqGHflY1daWc aaqaaiaaigdaaeaacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaeyyXIC TaamiEaaaacqGHflY1daWcaaqaaiaaigdaaeaacaWG5bWaaWbaaSqa beaacaaI1aaaaOGaeyyXICTaamyEamaaCaaaleqabaGaaG4maaaaaa GccqGHflY1daWcaaqaaiaadQhacqGHflY1caWG6bWaaWbaaSqabeaa caaI1aaaaaGcbaGaaGymaaaaaaa@5256@

Putting it all together:

2 x 2 y 5 z 2 2 x y 3 z 5 = 2 3 z 6 x 3 y 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIYaGaamiEamaaCaaaleqabaGaeyOeI0IaaGOmaaaakiaadMhadaah aaWcbeqaaiabgkHiTiaaiwdaaaGccaWG6baabaGaaGOmamaaCaaale qabaGaeyOeI0IaaGOmaaaakiaadIhacaWG5bWaaWbaaSqabeaacaaI ZaaaaOGaamOEamaaCaaaleqabaGaeyOeI0IaaGynaaaaaaGccqGH9a qpdaWcaaqaaiaaikdadaahaaWcbeqaaiaaiodaaaGccaWG6bWaaWba aSqabeaacaaI2aaaaaGcbaGaamiEamaaCaaaleqabaGaaG4maaaaki aadMhadaahaaWcbeqaaiaaiIdaaaaaaaaa@4E87@ .

Short-Cut 4: Negative exponents are division, so they need to be rewritten as multiplication by writing the reciprocal and changing the sign of the exponent. The last common question is what happens to the negative sign for the reciprocal? What happens to the division sign here: 3÷5=3× 1 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiabgE pa4kaaiwdacqGH9aqpcaaIZaGaey41aq7aaSaaaeaacaaIXaaabaGa aGynaaaaaaa@3F12@ . When you rewrite division you are writing it as multiplication. Positive exponents are repeated multiplication.

a m = 1 a m , 1 a m = a m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaeyOeI0IaamyBaaaakiabg2da9maalaaabaGaaGymaaqa aiaadggadaahaaWcbeqaaiaad2gaaaaaaOGaaiilaiaaykW7caaMc8 UaaGPaVlaaykW7daWcaaqaaiaaigdaaeaacaWGHbWaaWbaaSqabeaa cqGHsislcaWGTbaaaaaakiabg2da9iaadggadaahaaWcbeqaaiaad2 gaaaaaaa@4A81@

What about Zero?

This is the second to last thing we need to learn about exponents. However, a lot of practice is required to master them fully.

To see why anything to the power of zero is one, let’s consider:

3 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGynaaaaaaa@37A0@

This equals three times itself five total times:

3 5 =3"#x22C5;3333 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGynaaaakiabg2da9iaaiodacqGHflY1caaIZaGaeyyX ICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaaaa@4589@

Now let’s divide this by 35.

3 5 3 5 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGynaaaaaaaaaa@3962@

Without using short-cut 3, we have this:

3 5 3 5 = 33333 33333 =1 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGynaaaaaaGccqGH9aqpdaWcaaqaaiaaiodacqGHflY1caaIZaGaey yXICTaaG4maiabgwSixlaaiodacqGHflY1caaIZaaabaGaaG4maiab gwSixlaaiodacqGHflY1caaIZaGaeyyXICTaaG4maiabgwSixlaaio daaaGaeyypa0JaaGymaaaa@55F5@

Using short-cut 3, we have this:

3 5 3 5 = 3 55 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aIZaWaaWbaaSqabeaacaaI1aaaaaGcbaGaaG4mamaaCaaaleqabaGa aGynaaaaaaGccqGH9aqpcaaIZaWaaWbaaSqabeaacaaI1aGaeyOeI0 IaaGynaaaaaaa@3DC7@

Five minutes five is zero:

3 55 = 3 0 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGynaiabgkHiTiaaiwdaaaGccqGH9aqpcaaIZaWaaWba aSqabeaacaaIWaaaaaaa@3BFF@

Then 30 = 1.

Τhe 3 was an arbitrary base. This would work with any number except zero. You cannot divide by zero, it does not give us a number.

The beautiful thing about this is that no matter how ugly the base is, if the exponent is zero, the answer is just one. No need to simplify or perform calculation.

( 3 2x1 e πi n=1 1 n 2 ) 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaada WcaaqaaiaaiodadaahaaWcbeqaaiaaikdacaWG4bGaeyOeI0IaaGym aaaakiabgwSixlaadwgadaahaaWcbeqaaiabec8aWjaadMgaaaaake aadaaeWbqaamaalaaabaGaaGymaaqaaiaad6gadaahaaWcbeqaaiaa ikdaaaaaaaqaaiaad6gacqGH9aqpcaaIXaaabaGaeyOhIukaniabgg HiLdaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIWaaaaOGaeyyp a0JaaGymaaaa@4DBA@

Let’s take a quick look at all of our rules so far.

Short-Cut

Example

a m a n = a m+n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgwSixlaadggadaahaaWcbeqaaiaad6ga aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaa aaaaa@4140@

5 8 5= 5 8+1 = 5 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGioaaaakiabgwSixlaaiwdacqGH9aqpcaaI1aWaaWba aSqabeaacaaI4aGaey4kaSIaaGymaaaakiabg2da9iaaiwdadaahaa WcbeqaaiaaiMdaaaaaaa@41C8@

( a m ) n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaWGUbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiaad6 gaaaaaaa@3EB7@

( 7 2 ) 5 = 7 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI3aWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaI1aaaaOGaeyypa0JaaG4namaaCaaaleqabaGaaGymaiaaic daaaaaaa@3D93@

a m a n = a mn MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca WGHbWaaWbaaSqabeaacaWGTbaaaaGcbaGaamyyamaaCaaaleqabaGa amOBaaaaaaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaeyOeI0 IaamOBaaaaaaa@3F11@

5 7 5 2 = 5 72 = 5 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aWaaWbaaSqabeaacaaI3aaaaaGcbaGaaGynamaaCaaaleqabaGa aGOmaaaaaaGccqGH9aqpcaaI1aWaaWbaaSqabeaacaaI3aGaeyOeI0 IaaGOmaaaakiabg2da9iaaiwdadaahaaWcbeqaaiaaiwdaaaaaaa@4087@

a m = 1 a m  &   1 a m = a m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaeyOeI0IaamyBaaaakiabg2da9maalaaabaGaaGymaaqa aiaadggadaahaaWcbeqaaiaad2gaaaaaaOGaaeiiaiaabAcacaqGGa GaaeiiamaalaaabaGaaGymaaqaaiaadggadaahaaWcbeqaaiabgkHi Tiaad2gaaaaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaaaaaa a@4637@

4 3 = 1 4 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaeyOeI0IaaG4maaaakiabg2da9maalaaabaGaaGymaaqa aiaaisdadaahaaWcbeqaaiaaiodaaaaaaaaa@3C0F@

a 0 =1 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGimaaaakiabg2da9iaaigdaaaa@398F@

50 = 1

 

Let’s try some practice problems.

Instructions: Simplify the following.

1. ( 2 8 ) 1/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaWbaaSqabeaacaaI4aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIXaGaai4laiaaiodaaaaaaa@3B8D@ 2. 3 x 2 ( 3 x 2 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaadI hadaahaaWcbeqaaiaaikdaaaGccqGHflY1daqadaqaaiaaiodacaWG 4bWaaWbaaSqabeaacaaIYaaaaaGccaGLOaGaayzkaaWaaWbaaSqabe aacaaIZaaaaaaa@400E@

 

 

3. 5 5 m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aaabaGaaGynamaaCaaaleqabaGaamyBaaaaaaaaaa@38A4@ 4. 5 2 x 3 y 5 5 3 x 4 y 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaaca aI1aWaaWbaaSqabeaacaaIYaaaaOGaamiEamaaCaaaleqabaGaeyOe I0IaaG4maaaakiaadMhadaahaaWcbeqaaiaaiwdaaaaakeaacaaI1a WaaWbaaSqabeaacqGHsislcaaIZaaaaOGaamiEamaaCaaaleqabaGa eyOeI0IaaGinaaaakiaadMhadaahaaWcbeqaaiabgkHiTiaaiwdaaa aaaaaa@44E1@



 

5. 7÷7÷7÷7÷7÷7÷7÷7 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4naiabgE pa4kaaiEdacqGH3daUcaaI3aGaey49aGRaaG4naiabgEpa4kaaiEda cqGH3daUcaaI3aGaey49aGRaaG4naiabgEpa4kaaiEdaaaa@4B9C@ 6. 9 x 2 y÷9 x 2 y MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaey49aGRaaGyoaiaadIha daahaaWcbeqaaiaaikdaaaGccaWG5baaaa@3F94@

 

 

 

 

7. 9 x 2 y÷( 9 x 2 y ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGyoaiaadI hadaahaaWcbeqaaiaaikdaaaGccaWG5bGaey49aG7aaeWaaeaacaaI 5aGaamiEamaaCaaaleqabaGaaGOmaaaakiaadMhaaiaawIcacaGLPa aaaaa@411D@ 8. ( x 2 2 x 6 ) 2 ( x 2 2 x 6 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bWaaWbaaSqabeaacaaIYaaaaOGaeyyXICTaaGOmaiaadIhadaah aaWcbeqaaiaaiAdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaik daaaGccqGHflY1daqadaqaaiaadIhadaahaaWcbeqaaiaaikdaaaGc cqGHflY1caaIYaGaamiEamaaCaaaleqabaGaaGOnaaaaaOGaayjkai aawMcaamaaCaaaleqabaGaeyOeI0IaaGOmaaaaaaa@4BF0@

 

 

9. ( a m ) n a m MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaWGUbaaaOGaeyyXICTaamyyamaaCaaaleqabaGaamyBaaaaaa a@3F08@ 10. ( 3 x 2 +4 ) 2 ( 3 x 2 +4 ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaSaaaeaada qadaqaaiaaiodacaWG4bWaaWbaaSqabeaacaaIYaaaaOGaey4kaSIa aGinaaGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaaaaaOqaamaabm aabaGaaG4maiaadIhadaahaaWcbeqaaiaaikdaaaGccqGHRaWkcaaI 0aaacaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaaaaaaaaaa@4390@

Exponents Part 1

exponents part 1

Exponents Part 1

Reading Math

One of the biggest things to understand about math is how it is written. The spatial arrangement of characters is syntax. Syntax, in English, refers to the arrangements of words to convey meaning.

Exponents are just a way of writing repeated multiplication. If we are multiplying a number by itself repeatedly, we can use an exponent to tell how many times the number is being multiplied. That’s it. Nothing tricky exists with exponents, no new operations or concepts to tackle. If you’re familiar with multiplication and its properties, exponents should be accessible.

That said, it is not without its pitfalls. A balance between conceptual understanding and procedural short-cuts is needed to avoid those pitfalls. The only way to strike that balance is through a careful progression of exercises and examples. An answer-getting mentality will lead to big troubles with exponents. People wishing to learn how exponents work must seek understanding.

Let’s establish some facts that will come into play with this first part of exponents.

1.      Exponents are repeated multiplication

2.      Multiplication is repeated addition

3.      Addition is “skip” counting

To simplify simple expressions with exponents you only need to know a few short-cuts, but to recall and understand, we need more. These facts are important.

With an exponential expression we have a base, the number being multiplied by itself, and the exponent, the small number on the top right of the base which describes how many times the base is being multiplied by itself.

a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGynaaaaaaa@37C9@

The number a is the base. We don’t know what a is other than it is a number. It’s not a big deal that we don’t know exactly what number it is, we still know things about this expression.

Five is the exponent, which means there are five a’s, all multiplying together, like this: aaaaa. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgw SixlaadggacqGHflY1caWGHbGaeyyXICTaamyyaiabgwSixlaadgga caGGUaaaaa@444F@

Something to keep in mind is that this expression equals another number. Since we don’t know what a is, we cannot find out exactly what it is, but we do know it’s a perfect 5th power number, like 32. See, 25 = 32.

What if we had another number multiplying with a5, like this:

a 5 b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGynaaaakiabgwSixlaadkgadaahaaWcbeqaaiaaioda aaaaaa@3BEE@

If we write this out, without the exponents we see we have 5 a’s and 3 b’s, all multiplying together. We don’t know what a or b equals, but we do know they’re multiplying so we could change the order of multiplication (commutative property) or group them together in anyway we wish (associative property) without changing the value.

a 5 b 3 =aaaaabbb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGynaaaakiabgwSixlaadkgadaahaaWcbeqaaiaaioda aaGccqGH9aqpcaWGHbGaeyyXICTaamyyaiabgwSixlaadggacqGHfl Y1caWGHbGaeyyXICTaamyyaiabgwSixlaadkgacqGHflY1caWGIbGa eyyXICTaamOyaaaa@5437@

And these would be the same:

( aa )( aaa )( bbb ) MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaeyyXICTaamyyaaGaayjkaiaawMcaamaabmaabaGaamyyaiab gwSixlaadggacqGHflY1caWGHbaacaGLOaGaayzkaaWaaeWaaeaaca WGIbGaeyyXICTaamOyaiabgwSixlaadkgaaiaawIcacaGLPaaaaaa@4D37@

( aa )[ ( aaa )( bbb ) ] MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaeyyXICTaamyyaaGaayjkaiaawMcaamaadmaabaWaaeWaaeaa caWGHbGaeyyXICTaamyyaiabgwSixlaadggaaiaawIcacaGLPaaada qadaqaaiaadkgacqGHflY1caWGIbGaeyyXICTaamOyaaGaayjkaiaa wMcaaaGaay5waiaaw2faaaaa@4F29@

( aa ) [ ab ] 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaeyyXICTaamyyaaGaayjkaiaawMcaamaadmaabaGaamyyaiaa dkgaaiaawUfacaGLDbaadaahaaWcbeqaaiaaiodaaaaaaa@403F@

a 2 [ ab ] 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaGOmaaaakmaadmaabaGaamyyaiaadkgaaiaawUfacaGL DbaadaahaaWcbeqaaiaaiodaaaaaaa@3C79@

This is true because the brackets group together the a and b, making them both the base. The brackets put them together. The base is ab, and the exponent is 3. This means we have ab multiplied by itself three times.

Keep in mind, these are steps but exploring how exponents work to help you learn to read the math for the intended meaning behind the spatial arrangement of bases, parenthesis and exponents.

Now, the bracketed expression above is different than ab3, which is abbb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgw SixlaadkgacqGHflY1caWGIbGaeyyXICTaamOyaaaa@4070@ .

( ab ) 3 a b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaakiab gcMi5kaadggacaWGIbWaaWbaaSqabeaacaaIZaaaaaaa@3EBF@

Let’s expand these exponents and see why this is:

( ab ) 3 a b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaamOyaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4maaaakiab gcMi5kaadggacaWGIbWaaWbaaSqabeaacaaIZaaaaaaa@3EBF@

Write out the base ab times itself three times:

( ab )( ab )( ab )abbb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaamOyaaGaayjkaiaawMcaamaabmaabaGaamyyaiaadkgaaiaa wIcacaGLPaaadaqadaqaaiaadggacaWGIbaacaGLOaGaayzkaaGaey iyIKRaamyyaiabgwSixlaadkgacqGHflY1caWGIbGaeyyXICTaamOy aaaa@4C39@

The commutative property of multiplication allows us to rearrange the order in which we multiply the a’s and b’s.

aaabbbabbb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiabgw SixlaadggacqGHflY1caWGHbGaeyyXICTaamOyaiabgwSixlaadkga cqGHflY1caWGIbGaeyiyIKRaamyyaiabgwSixlaadkgacqGHflY1ca WGIbGaeyyXICTaamOyaaaa@5310@

Rewriting this repeated multiplication we get:

a 3 b 3 a b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaG4maaaakiaadkgadaahaaWcbeqaaiaaiodaaaGccqGH GjsUcaWGHbGaamOyamaaCaaaleqabaGaaG4maaaaaaa@3E2A@

The following, though, is true:

( a b 3 )=a b 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaamOyamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaaiab g2da9iaadggacaWGIbWaaWbaaSqabeaacaaIZaaaaaaa@3DFE@

On the right, the a has only an exponent of 1. If you do not see an exponent written, it is one. If we write it out we see:

( abbb )=abbb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbGaeyyXICTaamOyaiabgwSixlaadkgacqGHflY1caWGIbaacaGL OaGaayzkaaGaeyypa0JaamyyaiabgwSixlaadkgacqGHflY1caWGIb GaeyyXICTaamOyaaaa@4D78@

In summary of this first exploration, the base can be tricky to see. Parenthesis group things together. An exponent written outside the parenthesis creates all of the terms inside the parenthesis as the base. But if numbers are multiplying, but not grouped, and one has an exponent, the exponent only belongs to the number just below it on the left. For example, 4 x 3 , MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinaiaadI hadaahaaWcbeqaaiaaiodaaaGccaGGSaaaaa@3956@ the four has an exponent of just one, while the x is being cubed.

Consider: ( x+5 ) 3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bGaey4kaSIaaGynaaGaayjkaiaawMcaamaaCaaaleqabaGaaG4m aaaakiaac6caaaa@3BC4@ This means the base is x + 5 and it is multiplied by itself three times.

( x+5 ) 3 =( x+5 )( x+5 )( x+5 ) ( x+5 ) 3 x 3 + 5 3     MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaadaqada qaaiaadIhacqGHRaWkcaaI1aaacaGLOaGaayzkaaWaaWbaaSqabeaa caaIZaaaaOGaeyypa0ZaaeWaaeaacaWG4bGaey4kaSIaaGynaaGaay jkaiaawMcaamaabmaabaGaamiEaiabgUcaRiaaiwdaaiaawIcacaGL PaaadaqadaqaaiaadIhacqGHRaWkcaaI1aaacaGLOaGaayzkaaaaba WaaeWaaeaacaWG4bGaey4kaSIaaGynaaGaayjkaiaawMcaamaaCaaa leqabaGaaG4maaaakiabgcMi5kaadIhadaahaaWcbeqaaiaaiodaaa GccqGHRaWkcaaI1aWaaWbaaSqabeaacaaIZaaaaOGaaeiiaiaabcca caqGGaaaaaa@55E5@

Repeated Multiplication Allows Us Some Short-Cuts

Consider the expression:

a 3 × a 2 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeaaaaaa6di eB1vgapeGaamyya8aadaahaaWcbeqaaiaaiodaaaGccqGHxdaTqqa6 daaaaaGuLrgapiGaamyya8aadaahaaWcbeqaaiaaikdaaaGccaGGUa aaaa@4153@

If we wrote this out, we would have:

aaa×aa MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeaaaaaa6di eB1vgapeGaamyyaiabgwSixlaadggacqGHflY1caWGHbWdaiabgEna 0cbbOpaaaaaasvgza8GacaWGHbGaeyyXICTaamyyaaaa@483B@ .

(Note: In math we don’t use colors to differentiate between two things. A red a and a blue a are the same. These are colored to help us keep of track of what’s happening with each part of the expression.)

This is three a’s multiplying with another two a’s. That means there are five a’s multiplying.

a 3 × a 2 = a 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeaaaaaa6di eB1vgapeGaamyya8aadaahaaWcbeqaaiaaiodaaaGccqGHxdaTqqa6 daaaaaGuLrgapiGaamyya8aadaahaaWcbeqaaiaaikdaaaGccqGH9a qpcaWGHbWaaWbaaSqabeaacaaI1aaaaaaa@4379@

Before we generalize this to find the short-cut, let us see something similar, but is a potential pitfall.

a 3 × b 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeaaaaaa6di eB1vgapeGaamyya8aadaahaaWcbeqaaiaaiodaaaGccqGHxdaTqqa6 daaaaaGuLrgapiGaamOya8aadaahaaWcbeqaaiaaikdaaaaaaa@4098@

If we write this out we get:

aaa×bb MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaaeeaaaaaa6di eB1vgapeGaamyyaiabgwSixlaadggacqGHflY1caWGHbWdaiabgEna 0cbbOpaaaaaasvgza8GacaWGIbGaeyyXICTaamOyaaaa@483D@

This would not be an exponent of 5, in anyway. An exponent of five means the base is being multiplied by itself five times. Here we have an a as a base, and three of those multiplying, and a b as a base, and two of those multiplying. Not five of anything.

The common language is that if the bases are the same we can add the exponents. This is a hand short-cut, but if you forget where it comes from and why it is true, you’ll undoubtedly confuse it with some of the other short-cuts that follow.

Short-Cut 1: If the bases are the same you can add the exponents. This is true because exponents are repeated multiplication and the associative property says that the order in which you group things does not matter (when multiplying).

a m × a n = a m+n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaamyBaaaakiabgEna0kaadggadaahaaWcbeqaaiaad6ga aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaWGTbGaey4kaSIaamOBaa aaaaa@410D@

The second short-cut comes from groups and exponents.

( a 3 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaaaaaaa@3A43@

This means the base is a3, and it is being multiplied by itself.

a 3 × a 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaG4maaaakiabgEna0kaadggadaahaaWcbeqaaiaaioda aaaaaa@3BB8@

Our previous short cut said that if the bases are the same, we can add the exponents because we are just adding how many of the base is being multiplied by itself.

a 3 × a 3 = a 3+3 = a 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaaCa aaleqabaGaaG4maaaakiabgEna0kaadggadaahaaWcbeqaaiaaioda aaGccqGH9aqpcaWGHbWaaWbaaSqabeaacaaIZaGaey4kaSIaaG4maa aakiabg2da9iaadggadaahaaWcbeqaaiaaiAdaaaaaaa@431A@

But this is not much of a short cut. Let us look at the original expression and the outcome and look for a pattern.

( a 3 ) 2 = a 6 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaaaaOGaeyypa0JaamyyamaaCaaaleqabaGaaGOnaaaaaa a@3D26@

Short-Cut 2: A power raised to another is multiplied.

( a m ) n = a m×n MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WGHbWaaWbaaSqabeaacaWGTbaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaWGUbaaaOGaeyypa0JaamyyamaaCaaaleqabaGaamyBaiabgE na0kaad6gaaaaaaa@40CE@

Be careful here, though:

a ( b 3 c 2 ) 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyamaabm aabaGaamOyamaaCaaaleqabaGaaG4maaaakiaadogadaahaaWcbeqa aiaaikdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaaaaa@3D08@ = a b 15 c 10 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyyaiaadk gadaahaaWcbeqaaiaaigdacaaI1aaaaOGaam4yamaaCaaaleqabaGa aGymaiaaicdaaaaaaa@3BFF@

Practice Problems

1.       x 4 x 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaGinaaaakiabgwSixlaadIhadaahaaWcbeqaaiaaikda aaaaaa@3C18@

 

 

 

 

8. ( 5xy ) 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aGaamiEaiaadMhaaiaawIcacaGLPaaadaahaaWcbeqaaiaaioda aaaaaa@3B23@

2.       y 9 y MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamyEamaaCa aaleqabaGaaGyoaaaakiabgwSixlaadMhaaaa@3B36@

 

 

 

 

9. ( 8 m 4 ) 2 m 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI4aGaamyBamaaCaaaleqabaGaaGinaaaaaOGaayjkaiaawMcaamaa CaaaleqabaGaaGOmaaaakiabgwSixlaad2gadaahaaWcbeqaaiaaio daaaaaaa@3F41@

3.       z 2 z z 3 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamOEamaaCa aaleqabaGaaGOmaaaakiabgwSixlaadQhacqGHflY1caWG6bWaaWba aSqabeaacaaIZaaaaaaa@3F64@

 

 

 

 

10. ( 3 x 5 ) 3 ( 3 2 x 7 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaGaamiEamaaCaaaleqabaGaaGynaaaaaOGaayjkaiaawMcaamaa CaaaleqabaGaaG4maaaakmaabmaabaGaaG4mamaaCaaaleqabaGaaG OmaaaakiaadIhadaahaaWcbeqaaiaaiEdaaaaakiaawIcacaGLPaaa daahaaWcbeqaaiaaikdaaaaaaa@413A@

4.       ( x 5 ) 2 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG4bWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaaaaaaa@3A5B@

 

 

 

 

11. 7 ( 7 2 x 4 ) 5 7 3 x 5 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4namaabm aabaGaaG4namaaCaaaleqabaGaaGOmaaaakiaadIhadaahaaWcbeqa aiaaisdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaaiwdaaaGccq GHflY1caaI3aWaaWbaaSqabeaacaaIZaaaaOGaamiEamaaCaaaleqa baGaaGynaaaaaaa@42C5@

5.       ( y 4 ) 6 MathType@MTEF@5@5@+= feaagKart1ev2aaatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca WG5bWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaI2aaaaaaa@3A5F@

 

 

 

 

 

12. 5 3 + 5 3 + 5 3 + 5 3 + 5 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaG4maaaakiabgUcaRiaaiwdadaahaaWcbeqaaiaaioda aaGccqGHRaWkcaaI1aWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaaG ynamaaCaaaleqabaGaaG4maaaakiabgUcaRiaaiwdadaahaaWcbeqa aiaaiodaaaaaaa@41F4@

6.       x 3 + x 3 + x 3 + x 8 + x 8 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaamiEamaaCa aaleqabaGaaG4maaaakiabgUcaRiaadIhadaahaaWcbeqaaiaaioda aaGccqGHRaWkcaWG4bWaaWbaaSqabeaacaaIZaaaaOGaey4kaSIaam iEamaaCaaaleqabaGaaGioaaaakiabgUcaRiaadIhadaahaaWcbeqa aiaaiIdaaaaaaa@4334@

 

 

 

13. 3 2 9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4mamaaCa aaleqabaGaaGOmaaaakiabgwSixlaaiMdaaaa@3AB4@

 

 

 

 

 

7.       4 x + 4 x + 4 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaamiEaaaakiabgUcaRiaaisdadaahaaWcbeqaaiaadIha aaGccqGHRaWkcaaI0aWaaWbaaSqabeaacaWG4baaaaaa@3D87@

 

 

 

14. 4 x 4 x 4 x MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGinamaaCa aaleqabaGaamiEaaaakiabgwSixlaaisdadaahaaWcbeqaaiaadIha aaGccqGHflY1caaI0aWaaWbaaSqabeaacaWG4baaaaaa@4057@

 

 

 

 

 

 

 

 

 

The Purpose of Homework and My Response

The purpose of homework is to promote learning.  That’s it.  It’s not a way to earn a grade or something to keep kids busy.  It’s also not something that just must be completed in order to stay out of trouble.  Homework is a chance to try things independently, make mistakes and explore the nature of those mistakes in order to better learn the material at hand.

If students are not learning from the homework, it is a waste of time and effort.  There are a few things that could cause students not to learn from the homework.  Even if the assignments are of high quality, without the reflection and correction piece, students will not learn much from homework.

Reflection and correction go together.  It’s not about getting right answers, but thinking about what caused mistakes, identifying misconceptions or procedural inefficiencies and replacing those.  To reflect a student should NOT erase their incorrect working but instead should write on their homework, in pen, what went wrong and what would have been better.

It is quite possible more can be learned when reviewing homework than any other time.  It is certainly a powerful experience.

Textbooks and videos, tutors and peer help offer little appropriate support to help make homework, or practice, meaningful.  Textbooks only provide correct answers, YouTube videos usually do similar treatment to topics as textbooks offer.

I wish to help students learn and believe that reviewing work that has been done is too powerful of an opportunity to pass.  The trick is, how can I provide reflection and insight when to someone I am not sitting with and talking to?  I think I can help provide this reflection piece by doing all of the practice problems myself on a document camera and discussing pitfalls and mistakes, as well as sharing my thinking about the problems as I tackle them.  Further, I can share typical mistakes I see from students as they are learning topics.

So as I develop the Algebra 1 content I will be working on adding videos and short written responses to the assignments to help students think about what they’ve done, its appropriateness, correctness and their level of understanding.