Intermediate Algebra

# 8.3Simplify Rational Exponents

Intermediate Algebra8.3 Simplify Rational Exponents

### Learning Objectives

By the end of this section, you will be able to:
• Simplify expressions with $a1na1n$
• Simplify expressions with $amnamn$
• Use the properties of exponents to simplify expressions with rational exponents

### Be Prepared 8.3

Before you get started, take this readiness quiz.

1. Add: $715+512.715+512.$
If you missed this problem, review Example 1.28.
2. Simplify: $(4x2y5)3.(4x2y5)3.$
If you missed this problem, review Example 5.18.
3. Simplify: $5−3.5−3.$
If you missed this problem, review Example 5.14.

### Simplify Expressions with $a1na1n$

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 $(am)n=am·n(am)n=am·n$ 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 $(8p)3=8.(8p)3=8.$ We will use the Power Property of Exponents to find the value of p.

$(8p)3=8Multiply the exponents on the left.83p=8Write the exponent 1 on the right.83p=81Since the bases are the same, the exponents must be equal.3p=1Solve forp.p=13(8p)3=8Multiply the exponents on the left.83p=8Write the exponent 1 on the right.83p=81Since the bases are the same, the exponents must be equal.3p=1Solve forp.p=13$

So $(813)3=8.(813)3=8.$ But we know also $(83)3=8.(83)3=8.$ Then it must be that $813=83.813=83.$

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

### Rational Exponent $a 1 n a 1 n$

If $anan$ is a real number and $n≥2,n≥2,$ then

$a1n=ana1n=an$

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 8.26

Write as a radical expression: $x12x12$ $y13y13$ $z14.z14.$

### Try It 8.51

Write as a radical expression: $t12t12$ $m13m13$ $r14.r14.$

### Try It 8.52

Write as a radial expression: $b16b16$ $z15z15$ $p14.p14.$

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 8.27

Write with a rational exponent: $5y5y$ $4x34x3$ $35z4.35z4.$

### Try It 8.53

Write with a rational exponent: $10m10m$ $3n53n5$ $36y4.36y4.$

### Try It 8.54

Write with a rational exponent: $3k73k7$ $5j45j4$ $82a3.82a3.$

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

### Example 8.28

Simplify: $25122512$ $64136413$ $25614.25614.$

### Try It 8.55

Simplify: $36123612$ $813813$ $1614.1614.$

### Try It 8.56

Simplify: $1001210012$ $27132713$ $8114.8114.$

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

### Example 8.29

Simplify: $(−16)14(−16)14$ $−1614−1614$ $(16)−14.(16)−14.$

### Try It 8.57

Simplify: $(−64)−12(−64)−12$ $−6412−6412$ $(64)−12.(64)−12.$

### Try It 8.58

Simplify: $(−256)14(−256)14$ $−25614−25614$ $(256)−14.(256)−14.$

### Simplify Expressions with $amnamn$

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

$amn=(a1n)m=(an)mandamn=(am)1n=amnamn=(a1n)m=(an)mandamn=(am)1n=amn$

This leads us to the following definition.

### Rational Exponent $a m n a m n$

For any positive integers m and n,

$amn=(an)mandamn=amnamn=(an)mandamn=amn$

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 8.30

Write with a rational exponent: $y3y3$ $(2x3)4(2x3)4$ $(3a4b)3.(3a4b)3.$

### Try It 8.59

Write with a rational exponent: $x5x5$ $(3y4)3(3y4)3$ $(2m3n)5.(2m3n)5.$

### Try It 8.60

Write with a rational exponent: $a25a25$ $(5ab3)5(5ab3)5$ $(7xyz)3.(7xyz)3.$

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

### Example 8.31

Simplify: $1252312523$ $16−3216−32$ $32−25.32−25.$

### Try It 8.61

Simplify: $27232723$ $81−3281−32$ $16−34.16−34.$

### Try It 8.62

Simplify: $432432$ $27−2327−23$ $625−34.625−34.$

### Example 8.32

Simplify: $−2532−2532$ $−25−32−25−32$ $(−25)32.(−25)32.$

### Try It 8.63

Simplify: $−1632−1632$ $−16−32−16−32$ $(−16)−32.(−16)−32.$

### Try It 8.64

Simplify: $−8132−8132$ $−81−32−81−32$ $(−81)−32.(−81)−32.$

### 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 Exponenets 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

$Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=am−n,a≠0Zero Exponent Definitiona0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0Negative Exponent Propertya−n=1an,a≠0Product Propertyam·an=am+nPower Property(am)n=am·nProduct to a Power(ab)m=ambmQuotient Propertyaman=am−n,a≠0Zero Exponent Definitiona0=1,a≠0Quotient to a Power Property(ab)m=ambm,b≠0Negative Exponent Propertya−n=1an,a≠0$

We will apply these properties in the next example.

### Example 8.33

Simplify: $x12·x56x12·x56$ $(z9)23(z9)23$ $x13x53.x13x53.$

### Try It 8.65

Simplify: $x16·x43x16·x43$ $(x6)43(x6)43$ $x23x53.x23x53.$

### Try It 8.66

Simplify: $y34·y58y34·y58$ $(m9)29(m9)29$ $d15d65.d15d65.$

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 8.34

Simplify: $(27u12)23(27u12)23$ $(m23n12)32.(m23n12)32.$

### Try It 8.67

Simplify: $(32x13)35(32x13)35$ $(x34y12)23.(x34y12)23.$

### Try It 8.68

Simplify: $(81n25)32(81n25)32$ $(a32b12)43.(a32b12)43.$

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

### Example 8.35

Simplify: $x34·x−14x−64x34·x−14x−64$ $(16x43y−56x−23y16)12.(16x43y−56x−23y16)12.$

### Try It 8.69

Simplify: $m23·m−13m−53m23·m−13m−53$ $(25m16n116m23n−16)12.(25m16n116m23n−16)12.$

### Try It 8.70

Simplify: $u45·u−25u−135u45·u−25u−135$ $(27x45y16x15y−56)13.(27x45y16x15y−56)13.$

### Media

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

### Section 8.3 Exercises

#### Practice Makes Perfect

Simplify expressions with $a1na1n$

In the following exercises, write as a radical expression.

119.

$x12x12$ $y13y13$ $z14z14$

120.

$r12r12$ $s13s13$ $t14t14$

121.

$u15u15$ $v19v19$ $w120w120$

122.

$g17g17$ $h15h15$ $j125j125$

In the following exercises, write with a rational exponent.

123.

$x7x7$ $y9y9$ $f5f5$

124.

$r8r8$ $s10s10$ $t4t4$

125.

$7c37c3$ $12d712d7$ $26b426b4$

126.

$5x45x4$ $9y89y8$ $73z573z5$

127.

$21p21p$ $8q48q4$ $436r6436r6$

128.

$25a325a3$ $3b3b$ $40c840c8$

In the following exercises, simplify.

129.

$81128112$ $1251312513$ $64126412$

130.

$6251462514$ $2431524315$ $32153215$

131.

$16141614$ $16121612$ $6251462514$

132.

$64136413$ $32153215$ $81148114$

133.

$(−216)13(−216)13$ $−21613−21613$ $(216)−13(216)−13$

134.

$(−1000)13(−1000)13$ $−100013−100013$ $(1000)−13(1000)−13$

135.

$(−81)14(−81)14$ $−8114−8114$ $(81)−14(81)−14$

136.

$(−49)12(−49)12$ $−4912−4912$ $(49)−12(49)−12$

137.

$(−36)12(−36)12$ $−3612−3612$ $(36)−12(36)−12$

138.

$(−16)14(−16)14$ $−1614−1614$ $16−1416−14$

139.

$(−100)12(−100)12$ $−10012−10012$ $(100)−12(100)−12$

140.

$(−32)15(−32)15$ $(243)−15(243)−15$ $−12513−12513$

Simplify Expressions with $amnamn$

In the following exercises, write with a rational exponent.

141.

$m5m5$ $(3y3)7(3y3)7$ $(4x5y)35(4x5y)35$

142.

$r74r74$ $(2pq5)3(2pq5)3$ $(12m7n)34(12m7n)34$

143.

$u25u25$ $(6x3)5(6x3)5$ $(18a5b)74(18a5b)74$

144.

$a3a3$ $(21v4)3(21v4)3$ $(2xy5z)24(2xy5z)24$

In the following exercises, simplify.

145.

$64526452$ $81−3281−32$ $(−27)23(−27)23$

146.

$25322532$ $9−329−32$ $(−64)23(−64)23$

147.

$32253225$ $27−2327−23$ $(−25)12(−25)12$

148.

$1003210032$ $49−5249−52$ $(−100)32(−100)32$

149.

$−932−932$ $−9−32−9−32$ $(−9)32(−9)32$

150.

$−6432−6432$ $−64−32−64−32$ $(−64)32(−64)32$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify. Assume all variables are positive.

151.

$c14·c58c14·c58$ $(p12)34(p12)34$ $r45r95r45r95$

152.

$652·612652·612$ $(b15)35(b15)35$ $w27w97w27w97$

153.

$y12·y34y12·y34$ $(x12)23(x12)23$ $m58m138m58m138$

154.

$q23·q56q23·q56$ $(h6)43(h6)43$ $n35n85n35n85$

155.

$(27q32)43(27q32)43$ $(a13b23)32(a13b23)32$

156.

$(64s37)16(64s37)16$ $(m43n12)34(m43n12)34$

157.

$(16u13)34(16u13)34$ $(4p13q12)32(4p13q12)32$

158.

$(625n83)34(625n83)34$ $(9x25y35)52(9x25y35)52$

159.

$r52·r−12r−32r52·r−12r−32$ $(36s15t−32s−95t12)12(36s15t−32s−95t12)12$

160.

$a34·a−14a−104a34·a−14a−104$ $(27b23c−52b−73c12)13(27b23c−52b−73c12)13$

161.

$c53·c−13c−23c53·c−13c−23$ $(8x53y−1227x−43y52)13(8x53y−1227x−43y52)13$

162.

$m74·m−54m−24m74·m−54m−24$ $(16m15n3281m95n−12)14(16m15n3281m95n−12)14$

#### Writing Exercises

163.

Show two different algebraic methods to simplify $432.432.$ Explain all your steps.

164.

Explain why the expression $(−16)32(−16)32$ cannot be evaluated.

#### Self Check

After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.

What does this checklist tell you about your mastery of this section? What steps will you take to improve?

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