Exponents Part 2

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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@