1 Chapter 3.2: Rational Exponents In the previous section, we learned that while positive exponents are repeated multiplication, negative exponents mean that we take the reciprocal, or divide.

Positive exponents repeated

Zero exponents one Negative exponents repeated

In this section, we look at what happens when we have an exponent that is a fraction. This is known as a rational exponent. The word Rational has the root word ratio in it; a rational exponent is the ratio of two whole numbers, also known as a fraction.

🙄🙄 😀😀 To find out what rational exponents do, let’s use our calculator to try out some examples. What does 25

12 give you on your

calculator? How about 3612 ? 100

12 ?

Calculator tip: To get 25

12 , use the decimal form of the fraction,

0.5

and the exponent key (^, yx or xy ): . Or Put parenthesis around the fraction:

25 (1/2) What answers do you get? What do you think fractional exponents are? You should get the following answers:

Why don’t mathematicians just say “fractional exponent,”

instead of “rational exponent”?

Maybe because the word ratio sounds smarter? Or maybe it’s just

traditional? Ratio is from Ancient Greek! That’s a pretty old tradition!

2

2512 = 5

3612 = 6

10012 = 10

The ½ power means the square root of x. In other words, 𝑥𝑥12 = √𝑥𝑥.

Thus, 2512 = √25 = 5

3612 = √36 = 6

10012 = √100 = 10

The square root is the opposite of squaring. For example, √25 = 5 because 52 = 25

The square root of a is the positive number we square to get a.

√36 = 6 because 62 = 36. Note that (−6)2 is also equal to 36, but the definition is only for the positive number we square to get a.

Another way to see why the square root and ½ power are the same is to look at the breakdown of factors in a number.

For example, the factors of 25 are 5•5, or 52

2512 = (5 ∙ 5)

12 = (52)

12

Next, remember the following property of exponents:

Thus, our next step is to multiply the exponents: 25

12 = (5 ∙ 5)

12 = (52)

12 = 52∙

12 2 ∙ 1

2= 2

1∙ 12

= 22

= 1

52∙12 = 51 = 5

Similarly, 3612 = (6 ∙ 6)

12 = (62)

12 = 62∙

12 = 61 = 6 😎😎

Notice that the ½ power and the 2nd power are opposites and cancel each other out.

(𝑎𝑎𝑥𝑥)𝑦𝑦 = 𝑎𝑎𝑥𝑥𝑦𝑦 𝑎𝑎≠ 0

This property tells us when the base is the same, and we are multiplying exponential expressions, we can combine exponents by multiplying them.

Reminder: When multiplying fractions, multiply straight across the tops, straight across the bottoms, then reduce. Or reduce as you go along. In the example, above, you could instead “cancel” the 2 on top and the 2 on the bottom to get 1.

3 What if we add variables to the mix? Let’s try some more examples, this time with both numbers and variables. EXAMPLES a. √49𝑎𝑎4𝑏𝑏10

We can begin by thinking about this problem in three separate parts. • Let’s start with √49. We know that is equal to 7, since 72 = 49. • Next, let’s try √𝑎𝑎4. For this part, it may be easier to think of the square root as the ½

power. √𝑎𝑎4 = (𝑎𝑎4)12. Using the property of exponents, (𝑎𝑎𝑥𝑥)𝑦𝑦 = 𝑎𝑎𝑥𝑥𝑦𝑦, we multiply

exponents: 4 ∙ 12

= 41∙ 12

= 42

= 2. Thus, (𝑎𝑎4)12 = 𝑎𝑎2. Note that we could also get this by

thinking that “half of 4 is 2.” • Finally, let’s try √𝑏𝑏10. Again, it may be easier to think of the square root as the ½

power. √𝑏𝑏10 = (𝑏𝑏10)12. We multiply exponents to get 10 ∙ 1

2= 10

1∙ 12

= 102

= 5. Note that we could also get this by thinking that “half of 10 is 5.” Thus, (𝑏𝑏10)

12 = 𝑏𝑏5.

Answer: √49𝑎𝑎4𝑏𝑏10 = 7𝑎𝑎2𝑏𝑏5

We can check this by squaring our result. If we take 7𝑎𝑎2𝑏𝑏5 and square it, we should get back 49𝑎𝑎4𝑏𝑏10

(7𝑎𝑎2𝑏𝑏5)2 = ?

Yes, if you square the above, you do get 49𝑎𝑎4𝑏𝑏10!

b. In this example, we’ll look at what happens if we have a number where we don’t know

the square root, and what happens if you have an odd exponent. �225𝑎𝑎5𝑏𝑏11

• Let’s start with √225. We could get this with a calculator, but what if this question was on the non-calculator part of a test? We can make a factor tree.

We know 225 is divisible by 5, since it ends in a 5.

225

If you work out the division by hand, you should get 225÷5=45

5 45 5 is prime (nothing can but 1 and 5 go into it), so stop at the 5.

45 is not prime, so find the factors of 45. Again, 5 is prime so we stop, but for 9, we find the factors. 9 5 The factors of 9 are 3 x 3. The prime factors of 225 are the bold

numbers in the factor tree. 3 3 5 × 3 × 3 × 5 = 52 ∙ 32

√225 = √52 ∙ 32 = (52 ∙ 32)12 = 5 ∙ 3 = 15

4

• Next, let’s try √𝑎𝑎5. Again, let’s use the ½ power. √𝑎𝑎5 = (𝑎𝑎5)12. We could multiply

exponents: 5 ∙ 12

= 41∙ 12

= 52

= 2 12 , but how do we interpret 2 ½ as an exponent?

Instead, let’s rewrite 𝑎𝑎5 as 𝑎𝑎4 ∙ 𝑎𝑎 Thus, (𝑎𝑎5)12 = (𝑎𝑎4 ∙ 𝑎𝑎)

12 = 𝑎𝑎2 ∙ 𝑎𝑎

12 = 𝑎𝑎2√𝑎𝑎.

Note that we could also get this by thinking that “half of 5 is 2, with one remaining inside the square root.” We can also think of this as 2 ½ as an exponent:

�𝑎𝑎5 = (𝑎𝑎5)12 = 𝑎𝑎2.5 = 𝑎𝑎2𝑎𝑎

12 = 𝑎𝑎2√𝑎𝑎

Here’s the 2nd power. Here’s the ½ power.

• Finally, let’s try √𝑏𝑏11. Again, we rewrite as √𝑏𝑏11 = (𝑏𝑏11)12. We multiply exponents to

get 11 ∙ 12

= 111∙ 12

= 112

= 5.5 𝑜𝑜𝑜𝑜 5 12. Thus, (𝑏𝑏11)

12 = 𝑏𝑏5√𝑏𝑏.

Here’s the 5th power. Here’s the ½ power.

Or, you could write √𝑏𝑏11 as √𝑏𝑏10𝑏𝑏 = (𝑏𝑏10 ∙ 𝑏𝑏)12 = 𝑏𝑏5 ∙ 𝑏𝑏

12 = 𝑏𝑏5√𝑏𝑏

Answer: √225𝑎𝑎5𝑏𝑏11 = 15𝑎𝑎2𝑏𝑏5√𝑎𝑎𝑏𝑏

c. The method, above, of square-rooting the “square-rootable” part and leaving theremainder behind inside the square root can also be applied to numbers. For example,√50 does not have an exact square root. But we can rewrite it as √50 = √5 ∙ 5 ∙ 2 =(5221)

12 = 5 ∙ 2

12 = 5√2

Now let’s look at other fractional exponents.

Let’s try more examples.

d. 6413 = √643 = 4 This is because 43 = 64

If you are trying this on your calculator, remember to put parenthesis around the fractional

exponent: 64 (1/3). Don’t use the decimal for 1/3, because you will have to round it off somewhere, since 1/3 = 0.333333…., so the rounded off version will be less accurate.

Definition

𝑥𝑥1𝑛𝑛 = √𝑥𝑥𝑛𝑛

In other words, raising x to the 1/n power is the same thing as taking the nth root of x.

5

e. (16)14 = √164 = 2 This is because (2)4 = 16

f. � 916�1/2

= � 916

= √9√16

= 34 This is because �3

4�2

= 34∙ 34

= 916

g. (−36)12 = √−36 This has no real-number solution, because there is no real number,

that when you square it, gives -36. For example, (−6)2 = 36,𝑛𝑛𝑜𝑜𝑛𝑛 − 36.

h. (−125𝑥𝑥3𝑦𝑦15)1/3

This one can be done in parts, using the definition as well as the exponent:

(−125)13 = √−1253 = −5

Next, (𝑥𝑥3)13 = 𝑥𝑥3∙

13 = 𝑥𝑥1 and (𝑦𝑦15)

13 = 𝑦𝑦15∙

13 = 𝑦𝑦5

Answer: (−125𝑥𝑥3𝑦𝑦15)1/3 = −5𝑥𝑥𝑦𝑦5

Next, let’s try some slightly more complicated rational exponents.

EXAMPLES i. 9

32 = √93

Order is unimportant on how you simplify these. You can either simplify the square root of 9 first, and then cube the expression, or you can cube 9, and then take the square root of the expression. Let’s use the first route, since it’s a little easier. √9 = 3, then raise this result to the third power: 33 = 27.

If we were to do it the other way, then 93 = 729 and √729 = 27.

See if you can try the remaining examples before turning the page.

j. 84/3

k. 253/2

l. 4−3/2

m. �278�−23

Definition 𝑥𝑥𝑚𝑚𝑛𝑛 = √𝑥𝑥𝑚𝑚𝑛𝑛

In other words, raising a term to a fractional power is the same thing as taking the nth root of that term, and then raising it to the mth power. The m (or the numerator) represents the exponent, and the n, or the denominator, represents the index, or root.

exponent 𝑥𝑥𝒎𝒎𝒏𝒏 = √𝑥𝑥𝒎𝒎𝒏𝒏

root

6

j. 84/3 = √843 . Here, it’s easier to do √83 first, which is 2. Next, raise 2 to the 4th power: 24 = 16

32

• 253/2 = √253 The square root of 25 is 5, then we have 53 = 125

• 4−3/2

For this problem, remember that the negative exponent means to take the reciprocal of the base. The base is 4, so we get 4−3/2 = 1 4Next, we translate the exponent of 3/2, remembering that 3 is the power and 2 is the root, so we have a square root: 43/2 = √43 = 23 = 8. Putting it all together, 4−3/2 = 1

√43= 1

23= 1

8

m. �278�−23

Again, the negative exponent means to take the reciprocal of the base. The base is 278

, so

the reciprocal of the base is 827

.

Next, the exponent of 2/3 means the power is 2 and the root is 3, so we have �� 827�23

.

We can then work on the numerator and denominator separately: √823 = 22 = 4 and √2723 = 32 = 9

Putting it all together, �278�−23 = � 8

27�23 = �� 8

27�23

= √823

√2723 = 49.

7

Simplifying variable expressions We can also simplify variable expressions with rational exponents, using the same properties of exponents we used in the previous section.

Property 1

Property 2

Property 3

Property 4

Property 5

𝑎𝑎𝑥𝑥 ∙ 𝑎𝑎𝑦𝑦 = 𝑎𝑎𝑥𝑥+𝑦𝑦 𝑎𝑎≠ 0

This property tells us when the base is the same, and we are multiplying exponential expressions, we can combine exponents by adding them.

𝑎𝑎𝑥𝑥

𝑎𝑎𝑦𝑦= 𝑎𝑎𝑥𝑥−𝑦𝑦 𝑎𝑎≠ 0

This property tells us when the base is the same, and we are dividing exponential expressions, we can combine exponents by subtracting them.

(𝑎𝑎𝑥𝑥)𝑦𝑦 = 𝑎𝑎𝑥𝑥𝑦𝑦 𝑎𝑎≠ 0

This property tells us when the base is the same, and we are multiplying exponential expressions, we can combine exponents by multiplying them.

𝑥𝑥0 = 1 (𝑥𝑥 ≠ 0)

Any (non-zero) number raised to the zero power is 1.

𝑥𝑥−𝑎𝑎 =1𝑥𝑥𝑎𝑎

𝑥𝑥 ≠ 0

A number x, to the exponent, -a, is the reciprocal of the same number raised to a.

Let’s now try some exercises with simplifying rational exponential expressions:

!a. !!!

!!

This is an example of variables of the same base being multiplied, so we must useProperty 3 and add the exponents.

!!+  !

!=   !

!"+   !

!"=   !"

!"    so  that  !

!! !

!!      =  !

!"!"

b.   !!!

!!

This is an example of a power raised to another power. Therefore, we must use property 5 andmultiply exponents.

!!x  !!  =   !

!"=   !

!  so  that !

!!

!!= !

!!

c.  !!!!

!!!

!!"!!!

We begin by simplifying the numerator.

!!!!!!!=  !!!"!

!!  (we  multiply  exponents  and  simplify)  =  !!!"!!.

Now, we have !!!"!!

!!"!!!  We can make the exponent on the r positive by bringing

it into the denominator, and we have:!!

!!  =   !!

!!!!"!!"! !!"!

!!

!!!= !!!

!!    and  2 −  !

!=   !

!− !

!= − !

!.    So,

!!

!"

!!

!=   !

!!"!!!

Combine the r's together by adding exponents (property 3).

Finally, combine the s's together by subtracting exponents(property 4).

Finally, make the exponent on the s positive by bringing the s into the numerator.

Offen

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