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Mathematics LibreTexts

5.1: Simplify Rational Exponents

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Learning Objectives

By the end of this section, you will be able to:

  • Simplify expressions with a1n
  • Simplify expressions with amn
  • Use the properties of exponents to simplify expressions with rational exponents
Warm Up

Compute or simplify

  1. 33
  2. 7674
  3. a19a14
  4. x2
  5. 3x3
Answer
  1. 27
  2. 72=49
  3. a33
  4. |x|
  5. x

A Note about nxn

We recall that in earlier sections that for any integer n2,

when the index n is odd nan=a

when the index n is even nan=|a|

We notice that if a>0, that is a positive real number, then nan=a for n2 no matter if n is even or odd. Since we will be dealing with some properties that may be confusing at first, we will treat all bases as positive real numbers.

Rationalizing the denominator.

No mathematician actually cares about rationalizing the denominator but it is a student learning objective for this class so... here we go.

Say we have a fraction 12 and we want to make the denominator into a rational, that is we want to rationalize the denominator. We know that 22=2 and multiplying by 1 doesn't change the number, so lets multiply by 22 since this is just a fancy way to write 1. We get 1222=222=22.

Exercise 5.1.1

Rationalize the denominator.

  1. 13
  2. 14
  3. 15
  4. 1x
Answer
  1. 33
  2. Trick question since 4=2 so 14=12
  3. 55
  4. xx

What about if we have a sum or difference in the denominator? For example we have 11+2, what do we do? Remember (x+y)(xy)=x2y2? we can use this to rationalize the denominator by using what is called the conjugate.

Definition: CONJUGATE

The conjugate of a+b is ab

and the conjugate of a+b is ab

So by multiplying by a sneaky 1, 1212, we can rationalize the denominator of 11+2. We get

Example 5.1.2

Rationalize the denominator of 11+2

Solution
Multiply top and bottom by the conjugate of the denominator =11+21212
Multiply =121222
Simplify

\(=\dfrac{1-\sqrt{2}}{1-2}

=\dfrac{1-\sqrt{2}}{-1}\

=1+\sqrt{2}\)

Let's do some examples

Exercise 5.1.3

Rationalize the denominator

  1. 11+3
  2. 15+4
  3. 125
  4. 13+x
  5. 1xy
Answer
  1. 1313=132
  2. 15+2=17
  3. 2+545=131=13
  4. 3x9x
  5. x+yx2y

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponents here to have them for reference as we simplify expressions.

Properties of Exponents

Properties of Exponents

If a and b are real numbers and m and n are rational numbers, then

Zero Exponent Property

a0=1,a0

Negative Exponent Property

an=1an,a0

Product Property

aman=am+n

Power Property

(am)n=amn

Product to a Power Property

(ab)m=ambm

Quotient Property

aman=amn,a0

Quotient to a Power Property

(ab)m=ambm,b0

Let's talk each property.

We saw the Zero Exponent Property in chapter 1 from dividing a1 by a.

We saw the Product Property in chapter 1 but a quick example would be 4243=44444=45

We saw the Negative Exponent Property in chapter 1 and it follows from dividing a0 by a to get a1.

To see the Power Property is true, let's consider an example. Say we have (22)3 and from the power property, this should be 26. Computationally, this is true since (22)3=43=64=26 but let's take a look at a picture for the intuition of why this is true.

2 to 6.png

We see that it takes 3 steps of 22 to go from 1=20 to 26. This makes sense since if you take 6 normal steps then this is the same as you taking 3 double steps since 32=6, and this is exactly what is happening in the picture above. For the same reason we see that if we take 2 triple steps then we took 6 normal steps, which is shown with the bottom arrows. So we get (an)m=anm=amn=(am)n.

The Product to a Power Property comes from noticing that multiplication is commutative, that is we can switch terms around without changing the product, i.e ab=ba. So let's take a look at (ab)2. Well, (ab)2=(ab)(ab)=abab=aabb=a2b2. The same thing goes for to the power of 3, (ab)3=ababab=aaabbb=a3b3. We can apply the same reasoning to the power of 4,5, 6, all the way up to any number n.

The Quotient Property is from the product property and negative exponent property which we saw in chapter 1. That is, aman=aman=am+(n)=amn

The Quotient to a Power Property follows from the negative exponent property and product to a power property. That is, (ab)2=(ab1)2=a2b2=a2b2

Use the Properties of Exponents to Simplify Expressions with Rational Exponents

The same properties of exponents that we have already used also apply to rational exponents. We will list the Properties of Exponents here to have them for reference as we simplify expressions.

Example 5.1.4

Simplify:

  1. x12x56
  2. (z9)23
  3. x13x53

Solution

a. The Product Property tells us that when we multiple the same base, we add the exponents.

x12x56

The bases are the same, so we add the exponents.

x12+56

Add the fractions.

x86

Simplify the exponent.

x43

b. The Power Property tells us that when we raise a power to a power, we multiple the exponents.

(z9)23

To raise a power to a power, we multiple the exponents.

z923

Simplify.

z6

c. The Quotient Property tells us that when we divide with the same base, we subtract the exponents.

x13x53

To divide with the same base, we subtract the exponents.

1x5313

Simplify.

1x43

Exercise 5.1.5

Simplify:

  1. x16x43
  2. (x6)43
  3. x23x53
Answer
  1. x32
  2. x8
  3. 1x
Exercise 5.1.6

Simplify:

  1. y34y58
  2. (m9)29
  3. d15d65
Answer
  1. y118
  2. m2
  3. 1d

Sometimes we need to use more than one property. In the next example, we will use both the Product to a Power Property and then the Power Property.

Example 5.1.7

Simplify:

  1. (27u12)23
  2. (m23n12)32

Solution:

a.

(27u12)23

First we use the Product to a Power Property.

(27)23(u12)23

Rewrite 27 as a power of 3.

(33)23(u12)23

To raise a power to a power, we multiple the exponents.

(32)(u13)

Simplify.

9u13

b.

(m23n12)32

First we use the Product to a Power Property.

(m23)32(n12)32

To raise a power to a power, we multiply the exponents.

mn34

Exercise 5.1.8

Simplify:

  1. (32x13)35
  2. (x34y12)23
Answer
  1. 8x15
  2. x12y13
Exercise 5.1.9

Simplify:

  1. (81n25)32
  2. (a32b12)43
Answer
  1. 729n35
  2. a2b23

We will use both the Product Property and the Quotient Property in the next example.

Example 5.1.10

Simplify:

  1. x34x14x64
  2. (16x43y56x23y16)12

Solution:

a.

x34x14x64

Use the Product Property in the numerator, add the exponents.

x24x64

Use the Quotient Property, subtract the exponents.

x84

Simplify.

x2

b.

(16x43y56x23y16)12

Use the Quotient Property, subtract the exponents.

(16x63y66)12

Simplify.

(16x2y)12

Use the Product to a Power Property, multiply the exponents.

4xy12

Exercise 5.1.11

Simplify:

  1. m23m13m53
  2. (25m16n116m23n16)12
Answer
  1. m2
  2. 5nm14
Exercise 5.1.12

Simplify:

  1. u45u25u135
  2. (27x45y16x15y56)13
Answer
  1. u3
  2. 3x15y13

Access these online resources for additional instruction and practice with simplifying rational exponents.

  • Review-Rational Exponents
  • Using Laws of Exponents on Radicals: Properties of Rational Exponents

Key Concepts

  • Rational Exponent a1n
    • If na is a real number and n2, then a1n=na.
  • Rational Exponent amn
    • For any positive integers m and n,
      amn=(na)m and amn=nam
  • Properties of Exponents
    • If a,b are real numbers and m,n are rational numbers, then
      • Product Property aman=am+n
      • Power Property (am)n=amn
      • Product to a Power (ab)m=ambm
      • Quotient Property aman=amn,a0
      • Zero Exponent Definition a0=1,a0
      • Quotient to a Power Property (ab)m=ambm,b0
      • Negative Exponent Property an=1an,a0

This page titled 5.1: Simplify Rational Exponents is shared under a CC BY-NC-SA 4.0 license and was authored, remixed, and/or curated by Stanislav A. Trunov and Elizabeth J. Hale via source content that was edited to the style and standards of the LibreTexts platform.

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