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2.19: Simplifying Rational Exponents

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Simplifying Expressions with a1n

Rational exponents are another way of writing expressions with radicals. When we use rational exponents, we can apply the properties of exponents to simplify expressions.

The Power Property for Exponents says that (am)n=amn when m and n are whole numbers. Let’s assume we are now not limited to whole numbers.

Suppose we want to find a number p such that (8p)3=8. We will use the Power Property of Exponents to find the value of p.

(8p)3=8

Multiple the exponents on the left.

83p=8

Write the exponent 1 on the right.

83p=81

Since the bases are the same, the exponents must be equal.

3p=1

Solve for p.

p=13

So (813)3=8. But we know also (38)3=8. Then it must be that 813=38.

This same logic can be used for any positive integer exponent n to show that a1n=na.

Rational Exponent a1n

If na is a real number and n2, then

a1n=na

The denominator of the rational exponent is the index of the radical.

There will be times when working with expressions will be easier if you use rational exponents and times when it will be easier if you use radicals. In the first few examples, you'll practice converting expressions between these two notations.

Example 2.19.1

Write as a radical expression:

  1. x12
  2. y13
  3. z14

Solution:

We want to write each expression in the form na.

a.

x12

The denominator of the rational exponent is 2, so the index of the radical is 2. We do not show the index when it is 2.

x

b.

y13

The denominator of the exponent is 3, so the index is 3.

3y

c.

z14

The denominator of the exponent is \4, so the index is 4.

4z

You Try 2.19.1

Write as a radical expression:

  1. t12
  2. m13
  3. r14
Answer
  1. t
  2. 3m
  3. 4r

In the next example, we will write each radical using a rational exponent. It is important to use parentheses around the entire expression in the radicand since the entire expression is raised to the rational power.

Example 2.19.2

Write with a rational exponent:

  1. 5y
  2. 34x
  3. 345z

Solution:

We want to write each radical in the form a1n

a.

5y

No index is shown, so it is 2.

The denominator of the exponent will be 2.

Put parentheses around the entire expression 5y.

(5y)12

b.

34x

The index is 3, so the denominator of the exponent is 3. Include parentheses (4x).

(4x)13

c.

345z

The index is 4, so the denominator of the exponent is 4. Put parentheses only around the 5z since 3 is not under the radical sign.

3(5z)14

You Try 2.19.2

Write with a rational exponent:

  1. 73k
  2. 45j
  3. 832a
Answer
  1. (3k)17
  2. (5j)14
  3. 8(2a)13

In the next example, you may find it easier to simplify the expressions if you rewrite them as radicals first.

Example 2.19.3

Simplify:

  1. 2512
  2. 6413
  3. 25614

Solution:

a.

2512

Rewrite as a square root.

25

Simplify.

5

b.

6413

Rewrite as a cube root.

364

Recognize 64 is a perfect cube.

343

Simplify.

4

c.

25614

Rewrite as a fourth root.

4256

Recognize 256 is a perfect fourth power.

444

Simplify.

4

You Try 2.19.3

Simplify:

  1. 3612
  2. 813
  3. 1614
Answer
  1. 6
  2. 2
  3. 2

Be careful of the placement of the negative signs in the next example. We will need to use the property an=1an in one case.

Example 2.19.4

Simplify:

  1. (16)14
  2. 1614
  3. (16)14

Solution:

a.

(16)14

Rewrite as a fourth root.

416

 

Simplify.

There is no real number raised to the fourth power that gives us -16 so this is not a real number.

b.

1614

The exponent only applies to the 16. Rewrite as a fourth root.

416

Rewrite 16 as 24

424

Simplify.

2

c.

(16)14

Rewrite using the property an=1an.

1(16)14

Rewrite as a fourth root.

1416

Rewrite 16 as 24.

1424

Simplify.

12

You Try 2.19.4

Simplify:

  1. (64)12
  2. 6412
  3. (64)12
Answer
  1. Not a real number
  2. 8
  3. 18

Simplifying Expressions with amn

We can look at amn in two ways. Remember the Power Property tells us to multiply the exponents and so (a1n)m and (am)1n both equal amn. If we write these expressions in radical form, we get

amn=(a1n)m=(na)m and amn=(am)1n=nam

This leads us to the following definition.

Rational Exponent amn

For any positive integers m and n,

amn=(na)m and amn=nam

Which form do we use to simplify an expression? We usually take the root first—that way we keep the numbers in the radicand smaller, before raising it to the power indicated.

Example 2.19.5

Write with a rational exponent:

  1. y3
  2. (32x)4
  3. (3a4b)3

Solution:

We want to use amn=nam to write each radical in the form amn

a.

Image showing the square root of y^3 with a rational exponent, y^(2/3)

Figure 2.19.1

b.

Image showing the cube root of 2x all raised to the fourth power with a rational exponent, (2x)^(4/3)

Figure 2.19.2

c.

Image showing the square root of quantity 3a divided by quantity 4b raised to the power of 3 with a rational exponent, ((3a)/(4b))^(3/2)

Figure 2.19.3

You Try 2.19.5

Write with a rational exponent:

  1. 5a2
  2. (35ab)5
  3. (7xyz)3
Answer
  1. a25
  2. (5ab)53
  3. (7xyz)32

Remember that an=1an. The negative sign in the exponent does not change the sign of the expression.

Example 2.19.6

Simplify:

  1. 12523
  2. 1632
  3. 3225

Solution:

We will rewrite the expression as a radical first using the defintion, amn=(na)m. This form lets us take the root first and so we keep the numbers in the radicand smaller than if we used the other form.

a.

12523

The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.

(3125)2

Simplify.

(5)2

25

b. We will rewrite each expression first using an=1an and then change to radical form.

1632

Rewrite using an=1an

11632

Change to radical form. The power of the radical is the numerator of the exponent, 3. The index is the denominator of the exponent, 2.

1(16)3

Simplify.

143

164

c.

3225

Rewrite using an=1an

13225

Change to radical form.

1(532)2

Rewrite the radicand as a power.

1(525)2

Simplify.

122

14

You Try 2.19.6

Simplify:

  1. 432
  2. 2723
  3. 62534
Answer
  1. 8
  2. 19
  3. 1125
Example 2.19.7

Simplify:

  1. 2532
  2. 2532
  3. (25)32

Solution:

a.

2532

Rewrite in radical form.

(25)3

Simplify the radical.

(5)3

Simplify.

125

b.

2532

Rewrite using an=1an.

(12532)

Rewrite in radical form.

(1(25)3)

Simplify the radical.

(1(5)3)

Simplify.

1125

c.

(25)32

Rewrite in radical form.

(25)3

There is no real number whose square root is 25.

Not a real number.

You Try 2.19.7

Simplify:

  1. 1632
  2. 1632
  3. (16)32
Answer
  1. 64
  2. 164
  3. Not a real number

Using 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

If a and b are real numbers and m and 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

We will apply these properties in the next example.

Example 2.19.8

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

You Try 2.19.8

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 2.19.9

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

You Try 2.19.9

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 2.19.10

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

You Try 2.19.10

Simplify:

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

This page titled 2.19: Simplifying Rational Exponents is shared under a CC BY 4.0 license and was authored, remixed, and/or curated by Katherine Skelton.

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