Rational Exponents and Logarithmic Counting …

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rational exponents

Rational Exponents

In the last section we looked at some expressions like, “What is the third root of twenty-seven, squared?” The math is kind of ugly looking.

27 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIYaGaaG4namaaCaaaleqabaGaaGOmaaaaaeaacaaIZaaaaaaa@392A@

The procedures are clunky and it is very easy to lose sight of the objective. What this expression is asking is what number cubed is twenty-seven squared. You could always square the 27, to arrive at 729 and see if that is a perfect cube.

There is a much more elegant way to go about this type of calculation. Turns out if we rewrite this expression with a rational exponent, life gets easier.

27 2 3 = 27 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIYaGaaG4namaaCaaaleqabaGaaGOmaaaaaeaacaaIZaaaaOGaeyyp a0JaaGOmaiaaiEdadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaa a@3E10@

These two statements are the same. They ask the same question, what number cubed is twenty-seven squared?

By now you should be familiar with perfect cubes and squares. Hopefully you’re also familiar with higher powers of 2 and 3, as well as a few others. For example, you should recognize that 625 is 5 4 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGinaaaakiaac6caaaa@385D@ If you don’t know that yet, a cheat sheet might be helpful.

Let’s look at our expression again. If you notice that 27 is a perfect cube, then you can rewrite it like this:

27 2/3 ( 3 3 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmaiaaiE dadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaakiabgkziUoaabmaa baGaaG4mamaaCaaaleqabaGaaG4maaaaaOGaayjkaiaawMcaamaaCa aaleqabaGaaGOmaiaac+cacaaIZaaaaaaa@4157@

Maybe you see what’s going to happen next, but if not, we have a power raised to another here, we can multiply those exponents. Three times two-thirds is two. This becomes three squared.

( 3 3 ) 2/3 3 2 =9 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIZaWaaWbaaSqabeaacaaIZaaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaGaai4laiaaiodaaaGccqGHsgIRcaaIZaWaaWbaaSqabe aacaaIYaaaaOGaeyypa0JaaGyoaaaa@40FA@

Not too bad! We factor, writing the base of twenty-seven as an exponent with a power that matches the denominator of the other exponent, multiply, reduce, done!

 

 

Let’s look at another.

Simplify:

625 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIZaGaai4laiaaisdaaaaaaa@3A8D@

We mentioned earlier that 625 was a power of 5, the fourth power of five. That’s the key to making these simple. Let’s rewrite 625 as a power of five.

( 5 4 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaaaaa@3B8F@

We can multiply those exponents, giving us five-cubed, or 125. Much cleaner than finding the fourth root of six hundred and twenty-five cubed.

What about something that doesn’t work out so, well, pretty? Something where the base cannot be rewritten as an exponent that matches the denominator?

32 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaG4maiaaik dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaaaaa@39CB@

This is where proficiency and familiarity with powers of two comes to play. Thirty-two is a power of two, just not the fourth power, but the fifth.

( 2 5 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaaaaa@3B8D@

If we multiplied these exponents together we end up with something that isn’t so pretty, 2 15/4 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaGymaiaaiwdacaGGVaGaaGinaaaakiaac6caaaa@3A87@ We could rewrite this by simplifying the exponent, but there’s a better way. Consider the following, and note that we broke the five twos into a group of four and another group of one.

( 2 5 ) 3/4 = ( 2 1 2 4 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaWbaaSqabeaacaaI1aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaGccqGH9aqpdaqadaqaaiaaikdada ahaaWcbeqaaiaaigdaaaGccqGHflY1caaIYaWaaWbaaSqabeaacaaI 0aaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIZaGaai4laiaais daaaaaaa@462A@

Now we’d have to multiply the exponents inside the parenthesis by ¾ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=5laaaa@384E@ , and will arrive at:

2 3/4 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maiaac+cacaaI0aaaaOGaeyyXICTaaGOmamaaCaaa leqabaGaaG4maaaaaaa@3D08@

Notice that 2 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOmamaaCa aaleqabaGaaG4maiaac+cacaaI0aaaaaaa@390E@ is irrational, so not much we can do with it, but two cubed is eight. Let’s write the rational number first, and rewrite that irrational number as a radical expression:

8 2 3 4 ,or8 8 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGioamaake aabaGaaGOmamaaCaaaleqabaGaaG4maaaaaeaacaaI0aaaaOGaaiil aiaaykW7caaMc8Uaam4BaiaadkhacaaMc8UaaGPaVlaaiIdadaGcba qaaiaaiIdaaSqaaiaaisdaaaaaaa@445B@ .

There’s an even easier way to think about these rational exponents. I'd like to introduce something called Logarithmic Counting.  For those who don't know what logarithms are, that might sound scary.

Do you remember learning how to multiply by 5s...how you'd skip count?  (5, 10, 15, 20, ...)  Logarithmic counting is the same way, except with exponents.  For example, by 2:  2, 4, 8, 16, 32, ... Well, what’s the fourth step of 2 when logarithmically counting? It’s 16, right? MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F6@

Let’s look at 16 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaaaaa@39CD@ . See the denominator of four? That means we’re looking for a fourth root, a number times itself four times that equals 16. The three, in the numerator, it says, what number is three of the four steps on the way to sixteen?

2 4 8 16

Above is how we get to sixteen by multiplying a number by itself four times. Do you see the third step is eight?

Let’s see how our procedure looks:

Procedure 1:

16 3/4 = ( 2 4 ) 3/4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaakiabg2da9maabmaa baGaaGOmamaaCaaaleqabaGaaGinaaaaaOGaayjkaiaawMcaamaaCa aaleqabaGaaG4maiaac+cacaaI0aaaaaaa@4072@

( 2 4 ) 3/4 = 2 4 1 × 3 4 = 2 3 ,or8. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aIYaWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIZaGaai4laiaaisdaaaGccqGH9aqpcaaIYaWaaWbaaSqabe aadaWcaaqaaiaaisdaaeaacaaIXaaaaiabgEna0oaalaaabaGaaG4m aaqaaiaaisdaaaaaaOGaeyypa0JaaGOmamaaCaaaleqabaGaaG4maa aakiaacYcacaaMc8UaaGPaVlaad+gacaWGYbGaaGPaVlaaykW7caaI 4aGaaiOlaaaa@4FAB@

Procedure 2:

16 3/4 = 16 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaiA dadaahaaWcbeqaaiaaiodacaGGVaGaaGinaaaakiabg2da9maakeaa baGaaGymaiaaiAdadaahaaWcbeqaaiaaiodaaaaabaGaaGinaaaaaa a@3E10@

16 3 4 = ( 2 4 ) 3 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIXaGaaGOnamaaCaaaleqabaGaaG4maaaaaeaacaaI0aaaaOGaeyyp a0ZaaOqaaeaadaqadaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaaki aawIcacaGLPaaadaahaaWcbeqaaiaaiodaaaaabaGaaGinaaaaaaa@3F2C@

( 2 4 ) 3 4 = 2 4 4 × 2 4 4 × 2 4 4 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaada qadaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaakiaawIcacaGLPaaa daahaaWcbeqaaiaaiodaaaaabaGaaGinaaaakiabg2da9maakeaaba GaaGOmamaaCaaaleqabaGaaGinaaaaaeaacaaI0aaaaOGaey41aq7a aOqaaeaacaaIYaWaaWbaaSqabeaacaaI0aaaaaqaaiaaisdaaaGccq GHxdaTdaGcbaqaaiaaikdadaahaaWcbeqaaiaaisdaaaaabaGaaGin aaaaaaa@479A@

2 4 4 × 2 4 4 × 2 4 4 =2×2×2,or8. MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aIYaWaaWbaaSqabeaacaaI0aaaaaqaaiaaisdaaaGccqGHxdaTdaGc baqaaiaaikdadaahaaWcbeqaaiaaisdaaaaabaGaaGinaaaakiabgE na0oaakeaabaGaaGOmamaaCaaaleqabaGaaGinaaaaaeaacaaI0aaa aOGaeyypa0JaaGOmaiabgEna0kaaikdacqGHxdaTcaaIYaGaaiilai aaykW7caaMc8Uaam4BaiaadkhacaaMc8UaaGPaVlaaykW7caaI4aGa aiOlaaaa@54D0@

The most elegant way is to realize the 16 is the fourth power of 2, and the fraction ¾ MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaacbaqcLbyaqa aaaaaaaaWdbiaa=5laaaa@384E@ is asking us for the third entry. What is 3/4s of the way to 16 when multiplying (exponents)?

Let’s look at 625 2/3 . MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccaGGUaaa aa@3B47@ Let’s do this three ways, first with radical notation, then by evaluating the base and simplifying the exponents, and then by thinking about what is two thirds of the way to 625.

Now this is going to be a tricky problem because 625 is NOT a perfect cube. It is the fourth power of 5, though, which means that 125 (which is five-cubed) times five is 625.

Radical Notation:

625 2/3 = 625 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp daGcbaqaaiaaiAdacaaIYaGaaGynamaaCaaaleqabaGaaGOmaaaaae aacaaIZaaaaaaa@3F8C@

625 2 3 = ( 5 4 ) 2 3 5 8 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI2aGaaGOmaiaaiwdadaahaaWcbeqaaiaaikdaaaaabaGaaG4maaaa kiabg2da9maakeaabaWaaeWaaeaacaaI1aWaaWbaaSqabeaacaaI0a aaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaaaaaqaaiaaioda aaGccqGHsgIRdaGcbaqaaiaaiwdadaahaaWcbeqaaiaaiIdaaaaaba GaaG4maaaaaaa@445D@

5 8 3 = 5 3 × 5 3 × 5 2 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI1aWaaWbaaSqabeaacaaI4aaaaaqaaiaaiodaaaGccqGH9aqpdaGc baqaaiaaiwdadaahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaI1aWaaW baaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqabaGaaGOm aaaaaeaacaaIZaaaaaaa@438B@

5 3 × 5 3 × 5 2 3 =5×5 25 3 ,or25 25 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaOqaaeaaca aI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqa baGaaG4maaaakiabgEna0kaaiwdadaahaaWcbeqaaiaaikdaaaaaba GaaG4maaaakiabg2da9iaaiwdacqGHxdaTcaaI1aWaaOqaaeaacaaI YaGaaGynaaWcbaGaaG4maaaakiaacYcacaaMc8UaaGPaVlaaykW7ca WGVbGaamOCaiaaykW7caaMc8UaaGPaVlaaikdacaaI1aWaaOqaaeaa caaIYaGaaGynaaWcbaGaaG4maaaaaaa@56AD@

Pretty ugly!

Exponential Notation:

625 2/3 = ( 5 4 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp daqadaqaaiaaiwdadaahaaWcbeqaaiaaisdaaaaakiaawIcacaGLPa aadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaa@4131@

( 5 4 ) 2/3 = ( 5 3 × 5 1 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aWaaWbaaSqabeaacaaI0aaaaaGccaGLOaGaayzkaaWaaWbaaSqa beaacaaIYaGaai4laiaaiodaaaGccqGH9aqpdaqadaqaaiaaiwdada ahaaWcbeqaaiaaiodaaaGccqGHxdaTcaaI1aWaaWbaaSqabeaacaaI XaaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaGaai4laiaaio daaaaaaa@45FA@

( 5 3 × 5 1 ) 2/3 = 5 2 × 5 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaWaaeWaaeaaca aI1aWaaWbaaSqabeaacaaIZaaaaOGaey41aqRaaGynamaaCaaaleqa baGaaGymaaaaaOGaayjkaiaawMcaamaaCaaaleqabaGaaGOmaiaac+ cacaaIZaaaaOGaeyypa0JaaGynamaaCaaaleqabaGaaGOmaaaakiab gEna0kaaiwdadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaa@4745@

5 2 × 5 2/3 =25× 5 2/3 ,or25 25 3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGynamaaCa aaleqabaGaaGOmaaaakiabgEna0kaaiwdadaahaaWcbeqaaiaaikda caGGVaGaaG4maaaakiabg2da9iaaikdacaaI1aGaey41aqRaaGynam aaCaaaleqabaGaaGOmaiaac+cacaaIZaaaaOGaaiilaiaaykW7caaM c8UaaGPaVlaad+gacaWGYbGaaGPaVlaaykW7caaMc8UaaGOmaiaaiw dadaGcbaqaaiaaikdacaaI1aaaleaacaaIZaaaaaaa@5447@

A little better, but still a few sticky points.

Now our third method.

625 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaaaaa@3A8B@ asks, “What is two thirds of the way to 625, for a cubed number?”

This 625 isn’t cubed, but a factor of it is.

625 2/3 = ( 125×5 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGOnaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqp daqadaqaaiaaigdacaaIYaGaaGynaiabgEna0kaaiwdaaiaawIcaca GLPaaadaahaaWcbeqaaiaaikdacaGGVaGaaG4maaaaaaa@4489@
This could also be written as:

125 2/3 × 5 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGHxdaT caaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaaaaa@3FBF@

I am certain that 5 to the two-thirds power is irrational because, well, five is a prime number. Let’s deal with the other portion.

The steps to 125 are: 5 25 125

The second step is 25.

125 2/3 × 5 2/3 =25× 5 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcbaGaaGymaiaaik dacaaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGHxdaT caaI1aWaaWbaaSqabeaacaaIYaGaai4laiaaiodaaaGccqGH9aqpca aIYaGaaGynaiabgEna0kaaiwdadaahaaWcbeqaaiaaikdacaGGVaGa aG4maaaaaaa@4779@

To summarize the denominator of the rational exponent is the index of a radical expression. The numerator is an exponent for the base. How you tackle the expressions is entirely up to you, but I would suggest proficiency in multiple methods as sometimes the math lends itself nicely to one method but not another.

 


Practice Problems:

MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGcaa@35F6@ Simplify the following: 1.   ( 16 x 16 ) 3/4 2. 128 5/6 3. 125 3 5 4. 32 3/5 5. ( 81 x 27 ) 2/3 MathType@MTEF@5@5@+= feaagKart1ev2aqatCvAUfeBSjuyZL2yd9gzLbvyNv2CaerbuLwBLn hiov2DGi1BTfMBaeXatLxBI9gBaerbd9wDYLwzYbItLDharqqtubsr 4rNCHbGeaGqiVu0Je9sqqrpepC0xbbL8F4rqqrFfpeea0xe9Lq=Jc9 vqaqpepm0xbba9pwe9Q8fs0=yqaqpepae9pg0FirpepeKkFr0xfr=x fr=xb9adbaqaaeGaciGaaiaabeqaamaabaabaaGceaqabeaacaqGtb GaaeyAaiaab2gacaqGWbGaaeiBaiaabMgacaqGMbGaaeyEaiaabcca caqG0bGaaeiAaiaabwgacaqGGaGaaeOzaiaab+gacaqGSbGaaeiBai aab+gacaqG3bGaaeyAaiaab6gacaqGNbGaaeOoaaqaaiaabgdacaqG UaGaaeiiaiaabccadaqadaqaaiaaigdacaaI2aGaamiEamaaCaaale qabaGaaGymaiaaiAdaaaaakiaawIcacaGLPaaadaahaaWcbeqaaiaa iodacaGGVaGaaGinaaaaaOqaaaqaaaqaaiaaikdacaGGUaGaaGPaVl aaykW7caaMc8UaaGymaiaaikdacaaI4aWaaWbaaSqabeaacaaI1aGa ai4laiaaiAdaaaaakeaaaeaaaeaaaeaacaaIZaGaaiOlaiaaykW7ca aMc8UaaGPaVpaakeaabaGaaGymaiaaikdacaaI1aWaaWbaaSqabeaa caaIZaaaaaqaaiaaiwdaaaaakeaaaeaaaeaaaeaacaaI0aGaaiOlai aaykW7caaMc8UaaGPaVlaaiodacaaIYaWaaWbaaSqabeaacaaIZaGa ai4laiaaiwdaaaaakeaaaeaaaeaaaeaacaaI1aGaaiOlaiaaykW7ca aMc8+aaeWaaeaacaaI4aGaaGymaiaadIhadaahaaWcbeqaaiaaikda caaI3aaaaaGccaGLOaGaayzkaaWaaWbaaSqabeaacaaIYaGaai4lai aaiodaaaaaaaa@80AF@

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