Elementary Algebra 2e · free preview

9.8 Rational Exponents

Learning objectives: Simplify expressions with ${a}^{\frac{1}{n}}$; Simplify expressions with ${a}^{\frac{m}{n}}$; Use the Laws of Exponents to simply expressions with rational exponents

1. Learning objectives

Learning objectives

Learning objectives

  • Simplify expressions with a1n{a}^{\frac{1}{n}}
  • Simplify expressions with amn{a}^{\frac{m}{n}}
  • Use the Laws of Exponents to simply expressions with rational exponents

2. Simplify Expressions with ${a}^{\frac{1}{n}}$

Simplify Expressions with a1n{a}^{\frac{1}{n}}

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{({a}^{m})}^{n}={a}^{m\cdot{}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{({8}^{p})}^{3}=8. We will use the Power Property of Exponents to find the value of p.

 (8p)3=8\,\begin{array}{rll} {({8}^{p})}^{3} & = & 8 \\ \end{array}
Multiply the exponents on the left. 83p=8\,\begin{array}{rll} {8}^{3p} & = & 8 \\ \end{array}
Write the exponent 1 on the right. 83p=81\,\begin{array}{rll} {8}^{3p} & = & {8}^{1} \\ \end{array}
The exponents must be equal. 3p=1\,\begin{array}{rll} 3p & = & 1 \\ \end{array}
Solve for p.p. p=13\,\begin{array}{rll} p & = & \frac{1}{3} \\ \end{array}
So (813)3=8.\begin{array}{rll} \mathrm{So}\,{({8}^{\frac{1}{3}})}^{3} & = & 8. \\ \end{array}
But we know also (83)3=8{(\sqrt[3]{8})}^{3}=8. Then it must be that 813=83{8}^{\frac{1}{3}}=\sqrt[3]{8}.

But we know also (83)3=8{(\sqrt[3]{8})}^{3}=8. Then it must be that 813=83{8}^{\frac{1}{3}}=\sqrt[3]{8}.

This same logic can be used for any positive integer exponent n to show that a1n=an{a}^{\frac{1}{n}}=\sqrt[n]{a}.

If an\sqrt[n]{a} is a real number and n≥2n\ge{}2, a1n=an{a}^{\frac{1}{n}}=\sqrt[n]{a}.

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.

3. Example 9.103

Example 9.103

Example 9.103

Write as a radical expression: ⓐ x12{x}^{\frac{1}{2}} ⓑ y13{y}^{\frac{1}{3}} ⓒ z14{z}^{\frac{1}{4}}.

Solution

We want to write each expression in the form an\sqrt[n]{a}.

ⓐ

x12{x}^{\frac{1}{2}}
The denominator of the exponent is 2, so the index of the radical is 2. We do not show the index when it is 2.x\sqrt{x}

ⓑ

y13{y}^{\frac{1}{3}}
The denominator of the exponent is 3, so the index is 3.y3\sqrt[3]{y}

ⓒ

z14{z}^{\frac{1}{4}}
The denominator of the exponent is 4, so the index is 4.z4\sqrt[4]{z}

4. Example 9.104

Example 9.104

Example 9.104

Write with a rational exponent: ⓐ x\sqrt{x} ⓑ y3\sqrt[3]{y} ⓒ z4\sqrt[4]{z}.

Solution

We want to write each radical in the form a1n{a}^{\frac{1}{n}}.

ⓐ

x\sqrt{x}
No index is shown, so it is 2. The denominator of the exponent will be 2.x12{x}^{\frac{1}{2}}

ⓑ

y3\sqrt[3]{y}
The index is 3, so the denominator of the exponent is 3.y13{y}^{\frac{1}{3}}

ⓒ

z4\sqrt[4]{z}
The index is 4, so the denominator of the exponent is 4.z14{z}^{\frac{1}{4}}

5. Example 9.105

Example 9.105

Example 9.105

Write with a rational exponent: ⓐ 5y\sqrt{5y} ⓑ 4x3\sqrt[3]{4x} ⓒ 35z43\sqrt[4]{5z}.

Solution

We want to write each radical in the form a1n{a}^{\frac{1}{n}}.

ⓐ

5y\sqrt{5y}
No index is shown, so it is 2. The denominator of the exponent will be 2.(5y)12{(5y)}^{\frac{1}{2}}

ⓑ

4x3\sqrt[3]{4x}
The index is 3, so the denominator of the exponent is 3.(4x)13{(4x)}^{\frac{1}{3}}

ⓒ

35z43\sqrt[4]{5z}
The index is 4, so the denominator of the exponent is 4.3(5z)143{(5z)}^{\frac{1}{4}}

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

6. Example 9.106

Example 9.106

Example 9.106

Simplify: ⓐ 2512{25}^{\frac{1}{2}} ⓑ 6413{64}^{\frac{1}{3}} ⓒ 25614{256}^{\frac{1}{4}}.

Solution

ⓐ

2512{25}^{\frac{1}{2}}
Rewrite as a square root.25\sqrt{25}
Simplify.55

ⓑ

6413{64}^{\frac{1}{3}}
Rewrite as a cube root.643\sqrt[3]{64}
Recognize 64 is a perfect cube.433\sqrt[3]{{4}^{3}}
Simplify.44

ⓒ

25614{256}^{\frac{1}{4}}
Rewrite as a fourth root.2564\sqrt[4]{256}
Recognize 256 is a perfect fourth power.444\sqrt[4]{{4}^{4}}
Simplify.44

Be careful of the placement of the negative signs in the next example. We will need to use the property a−n=1an{a}^{-n}=\frac{1}{{a}^{n}} in one case.

7. Example 9.107

Example 9.107

Example 9.107

Simplify: ⓐ (−64)13{(-64)}^{\frac{1}{3}} ⓑ −6413-{64}^{\frac{1}{3}} ⓒ (64)−13{(64)}^{-\frac{1}{3}}.

Solution

ⓐ

(−64)13{(-64)}^{\frac{1}{3}}
Rewrite as a cube root.−643\sqrt[3]{-64}
Rewrite −64-64 as a perfect cube.(−4)33\sqrt[3]{{(-4)}^{3}}
Simplify.−4-4

ⓑ

−6413-{64}^{\frac{1}{3}}
The exponent applies only to the 64.−(6413)-({64}^{\frac{1}{3}})
Rewrite as a cube root.−643-\sqrt[3]{64}
Rewrite 64 as 43.{4}^{3}.−433-\sqrt[3]{{4}^{3}}
Simplify.−4-4

ⓒ

(64)−13{(64)}^{-\frac{1}{3}}
Rewrite as a fraction with a positive exponent, using the property, a−n=1an.{a}^{-n}=\frac{1}{{a}^{n}}.1643\frac{1}{\sqrt[3]{64}}
Write as a cube root.
Rewrite 64 as 43.{4}^{3}.1433\frac{1}{\sqrt[3]{{4}^{3}}}
Simplify.14\frac{1}{4}

8. Example 9.108

Example 9.108

Example 9.108

Simplify: ⓐ (−16)14{(-16)}^{\frac{1}{4}} ⓑ −1614-{16}^{\frac{1}{4}} ⓒ (16)−14{(16)}^{-\frac{1}{4}}.

Solution

ⓐ

(−16)14{(-16)}^{\frac{1}{4}}
Rewrite as a fourth root.−164\sqrt[4]{-16}
There is no real number whose fourth power is −16.-16.

ⓑ

−1614-{16}^{\frac{1}{4}}
The exponent only applies to the 16. Rewrite as a fourth root.−164-\sqrt[4]{16}
Rewrite 1616 as 24.{2}^{4}.−244-\sqrt[4]{{2}^{4}}
Simplify.−2-2

ⓒ

(16)−14{(16)}^{-\frac{1}{4}}
Rewrite using the property a−n=1an.{a}^{-n}=\frac{1}{{a}^{n}}.1(16)14\frac{1}{{(16)}^{\frac{1}{4}}}
Rewrite as a fourth root.1164\frac{1}{\sqrt[4]{16}}
Rewrite 1616 as 24.{2}^{4}.1244\frac{1}{\sqrt[4]{{2}^{4}}}
Simplify.12\frac{1}{2}

9. Simplify Expressions with ${a}^{\frac{m}{n}}$

Simplify Expressions with amn{a}^{\frac{m}{n}}

Let’s work with the Power Property for Exponents some more.

Suppose we raise a1n{a}^{\frac{1}{n}} to the power m.

(a1n)m{({a}^{\frac{1}{n}})}^{m}
Multiply the exponents.a1n⋅m{a}^{\frac{1}{n}}{}^{\cdot{}m}
Simplify.amn{a}^{\frac{m}{n}}
So amn=(an)m.{a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m}.

Now suppose we take am{a}^{m} to the 1n\frac{1}{n} power.

(am)1n{({a}^{m})}^{\frac{1}{n}}
Multiply the exponents.am⋅1n{a}^{m\cdot\frac{1}{n}}
Simplify.amn{a}^{\frac{m}{n}}
So amn=amn{a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}} also.

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.

For any positive integers m and n,

amn=(an)mamn=amn\begin{array}{l} {a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{m} \\ {a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}} \\ \end{array}

10. Example 9.109

Example 9.109

Example 9.109

Write with a rational exponent: ⓐ y3\sqrt{{y}^{3}} ⓑ x23\sqrt[3]{{x}^{2}} ⓒ z34\sqrt[4]{{z}^{3}}.

Solution

We want to use amn=amn{a}^{\frac{m}{n}}=\sqrt[n]{{a}^{m}} to write each radical in the form amn{a}^{\frac{m}{n}}.

  1. ⓐ
  1. ⓑ
  1. ⓒ

11. Example 9.110

Example 9.110

Example 9.110

Simplify: ⓐ 932{9}^{\frac{3}{2}} ⓑ 12523{125}^{\frac{2}{3}} ⓒ 8134{81}^{\frac{3}{4}}.

Solution

We will rewrite each expression as a radical first using the property, amn=(an)m{a}^{\frac{m}{n}}={(\sqrt[n]{a})}^{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.

ⓐ

932{9}^{\frac{3}{2}}
The power of the radical is the numerator of the exponent, 3. Since the denominator of the exponent is 2, this is a square root.(9)3{(\sqrt{9})}^{3}
Simplify.(3)3{(3)}^{3}
2727

ⓑ

12523{125}^{\frac{2}{3}}
The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.(1253)2{(\sqrt[3]{125})}^{2}
Simplify.(5)2{(5)}^{2}
2525

ⓒ

8134{81}^{\frac{3}{4}}
The power of the radical is the numerator of the exponent, 3. The index of the radical is the denominator of the exponent, 4.(814)3{(\sqrt[4]{81})}^{3}
Simplify.(3)3{(3)}^{3}
2727

Remember that b−p=1bp{b}^{-p}=\frac{1}{{b}^{p}}. The negative sign in the exponent does not change the sign of the expression.

12. Example 9.111

Example 9.111

Example 9.111

Simplify: ⓐ 16−32{16}^{-\frac{3}{2}} ⓑ 32−25{32}^{-\frac{2}{5}} ⓒ 4−52{4}^{-\frac{5}{2}}.

Solution

We will rewrite each expression first using b−p=1bp{b}^{-p}=\frac{1}{{b}^{p}} and then change to radical form.

ⓐ

16−32{16}^{-\frac{3}{2}}
Rewrite using b−p=1bp.{b}^{-p}=\frac{1}{{b}^{p}}.11632\frac{1}{{16}^{\frac{3}{2}}}
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\frac{1}{{(\sqrt{16})}^{3}}
Simplify.143\frac{1}{{4}^{3}}
164\frac{1}{64}

ⓑ

32−25{32}^{-\frac{2}{5}}
Rewrite using b−p=1bp.{b}^{-p}=\frac{1}{{b}^{p}}.13225\frac{1}{{32}^{\frac{2}{5}}}
Change to radical form.1(325)2\frac{1}{{(\sqrt[5]{32})}^{2}}
Rewrite the radicand as a power.1(255)2\frac{1}{{(\sqrt[5]{{2}^{5}})}^{2}}
Simplify.122\frac{1}{{2}^{2}}
14\frac{1}{4}

ⓒ

4−52{4}^{-\frac{5}{2}}
Rewrite using b−p=1bp.{b}^{-p}=\frac{1}{{b}^{p}}.1452\frac{1}{{4}^{\frac{5}{2}}}
Change to radical form.1(4)5\frac{1}{{(\sqrt{4})}^{5}}
Simplify.125\frac{1}{{2}^{5}}
132\frac{1}{32}

13. Example 9.112

Example 9.112

Example 9.112

Simplify: ⓐ −2532-{25}^{\frac{3}{2}} ⓑ −25−32-{25}^{-\frac{3}{2}} ⓒ (−25)32{(-25)}^{\frac{3}{2}}.

Solution

ⓐ

−2532-{25}^{\frac{3}{2}}
Rewrite in radical form.−(25)3-{(\sqrt{25})}^{3}
Simplify the radical.−(5)3-{(5)}^{3}
Simplify.−125-125

ⓑ

−25−32-{25}^{-\frac{3}{2}}
Rewrite using b−p=1bp.{b}^{-p}=\frac{1}{{b}^{p}}.−(12532)-(\frac{1}{{25}^{\frac{3}{2}}})
Rewrite in radical form.−(1(25)3)-(\frac{1}{{(\sqrt{25})}^{3}})
Simplify the radical.−(1(5)3)-(\frac{1}{{(5)}^{3}})
Simplify.−1125-\frac{1}{125}

ⓒ

(−25)32{(-25)}^{\frac{3}{2}}
Rewrite in radical form.(−25)3{(\sqrt{-25})}^{3}
There is no real number whose square root is −25.-25.Not a real number.

14. Use the Laws of Exponents to Simplify Expressions with Rational Exponents

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

The same laws of exponents that we already used apply to rational exponents, too. We will list the Exponent Properties here to have them for reference as we simplify expressions.

If a,ba,b are real numbers and m,nm,n are rational numbers, then ProductPropertyam⋅an=am+nPowerProperty(am)n=am⋅nProducttoaPower(ab)m=ambmQuotientPropertyaman=am−n,  a≠0,  m>naman=1an−m,  a≠0,  n>mZeroExponentDefinitiona0=1,  a≠0QuotienttoaPowerProperty(ab)m=ambm,  b≠0\begin{array}{lccl} \mathrm{Product Property} & & & {a}^{m}\cdot{a}^{n}={a}^{m+n} \\ \mathrm{Power Property} & & & {({a}^{m})}^{n}={a}^{m\cdot{}n} \\ \mathrm{Product to a Power} & & & {(ab)}^{m}={a}^{m}{b}^{m} \\ \mathrm{Quotient Property} & & & \frac{{a}^{m}}{{a}^{n}}={a}^{m-n},\,\,a\ne{}0,\,\,m>n \\ & & & \frac{{a}^{m}}{{a}^{n}}=\frac{1}{{a}^{n-m}},\,\,a\ne{}0,\,\,n>m \\ \mathrm{Zero Exponent Definition} & & & {a}^{0}=1,\,\,a\ne{}0 \\ \mathrm{Quotient to a Power Property} & & & {(\frac{a}{b})}^{m}=\frac{{a}^{m}}{{b}^{m}},\,\,b\ne{}0 \\ \end{array}

When we multiply the same base, we add the exponents.

15. Example 9.113

Example 9.113

Example 9.113

Simplify: ⓐ 212⋅252{2}^{\frac{1}{2}}\cdot{2}^{\frac{5}{2}} ⓑ x23⋅x43{x}^{\frac{2}{3}}\cdot{x}^{\frac{4}{3}} ⓒ z34⋅z54{z}^{\frac{3}{4}}\cdot{z}^{\frac{5}{4}}.

Solution

ⓐ

212⋅252{2}^{\frac{1}{2}}\cdot{2}^{\frac{5}{2}}
The bases are the same, so we add the exponents.212+52{2}^{\frac{1}{2}+\frac{5}{2}}
Add the fractions.262{2}^{\frac{6}{2}}
Simplify the exponent.23{2}^{3}
Simplify.88

ⓑ

x23⋅x43{x}^{\frac{2}{3}}\cdot{x}^{\frac{4}{3}}
The bases are the same, so we add the exponents.x23+43{x}^{\frac{2}{3}+\frac{4}{3}}
Add the fractions.x63{x}^{\frac{6}{3}}
Simplify.x2{x}^{2}

ⓒ

z34⋅z54{z}^{\frac{3}{4}}\cdot{z}^{\frac{5}{4}}
The bases are the same, so we add the exponents.z34+54{z}^{\frac{3}{4}+\frac{5}{4}}
Add the fractions.z84{z}^{\frac{8}{4}}
Simplify.z2{z}^{2}

We will use the Power Property in the next example.

16. Example 9.114

Example 9.114

Example 9.114

Simplify: ⓐ (x4)12{({x}^{4})}^{\frac{1}{2}} ⓑ (y6)13{({y}^{6})}^{\frac{1}{3}} ⓒ (z9)23{({z}^{9})}^{\frac{2}{3}}.

Solution

ⓐ

(x4)12{({x}^{4})}^{\frac{1}{2}}
To raise a power to a power, we multiply the exponents.x4⋅12{x}^{4\cdot\frac{1}{2}}
Simplify.x2{x}^{2}

ⓑ

(y6)13{({y}^{6})}^{\frac{1}{3}}
To raise a power to a power, we multiply the exponents.y6⋅13{y}^{6\cdot\frac{1}{3}}
Simplify.y2{y}^{2}

ⓒ

(z9)23{({z}^{9})}^{\frac{2}{3}}
To raise a power to a power, we multiply the exponents.z9⋅23{z}^{9\cdot\frac{2}{3}}
Simplify.z6{z}^{6}

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

17. Example 9.115

Example 9.115

Example 9.115

Simplify: ⓐ x43x13\frac{{x}^{\frac{4}{3}}}{{x}^{\frac{1}{3}}} ⓑ y34y14\frac{{y}^{\frac{3}{4}}}{{y}^{\frac{1}{4}}} ⓒ z23z53\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{5}{3}}}.

Solution

ⓐ

x43x13\frac{{x}^{\frac{4}{3}}}{{x}^{\frac{1}{3}}}
To divide with the same base, we subtract the exponents.x43−13{x}^{\frac{4}{3}-\frac{1}{3}}
Simplify.xx

ⓑ

y34y14\frac{{y}^{\frac{3}{4}}}{{y}^{\frac{1}{4}}}
To divide with the same base, we subtract the exponents.y34−14{y}^{\frac{3}{4}-\frac{1}{4}}
Simplify.y12{y}^{\frac{1}{2}}

ⓒ

z23z53\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{5}{3}}}
To divide with the same base, we subtract the exponents.z23−53{z}^{\frac{2}{3}-\frac{5}{3}}
Rewrite without a negative exponent.1z\frac{1}{z}

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

18. Example 9.116

Example 9.116

Example 9.116

Simplify: ⓐ (27u12)23{(27{u}^{\frac{1}{2}})}^{\frac{2}{3}} ⓑ (8v14)23{(8{v}^{\frac{1}{4}})}^{\frac{2}{3}}.

Solution

ⓐ

(27u12)23{(27{u}^{\frac{1}{2}})}^{\frac{2}{3}}
First we use the Product to a Power Property.(27)23(u12)23{(27)}^{\frac{2}{3}}{({u}^{\frac{1}{2}})}^{\frac{2}{3}}
Rewrite 27 as a power of 3.(33)23(u12)23{({3}^{3})}^{\frac{2}{3}}{({u}^{\frac{1}{2}})}^{\frac{2}{3}}
To raise a power to a power, we multiply the exponents.(32)(u13)({3}^{2})({u}^{\frac{1}{3}})
Simplify.9u139{u}^{\frac{1}{3}}

ⓑ

(8v14)23{(8{v}^{\frac{1}{4}})}^{\frac{2}{3}}
First we use the Product to a Power Property.(8)23(v14)23{(8)}^{\frac{2}{3}}{({v}^{\frac{1}{4}})}^{\frac{2}{3}}
Rewrite 8 as a power of 2.(23)23(v14)23{({2}^{3})}^{\frac{2}{3}}{({v}^{\frac{1}{4}})}^{\frac{2}{3}}
To raise a power to a power, we multiply the exponents.(22)(v16)({2}^{2})({v}^{\frac{1}{6}})
Simplify.4v164{v}^{\frac{1}{6}}

19. Example 9.117

Example 9.117

Example 9.117

Simplify: ⓐ (m3n9)13{({m}^{3}{n}^{9})}^{\frac{1}{3}} ⓑ (p4q8)14{({p}^{4}{q}^{8})}^{\frac{1}{4}}.

Solution

ⓐ

(m3n9)13{({m}^{3}{n}^{9})}^{\frac{1}{3}}
First we use the Product to a Power Property.(m3)13(n9)13{({m}^{3})}^{\frac{1}{3}}{({n}^{9})}^{\frac{1}{3}}
To raise a power to a power, we multiply the exponents.mn3m{n}^{3}

ⓑ

(p4q8)14{({p}^{4}{q}^{8})}^{\frac{1}{4}}
First we use the Product to a Power Property.(p4)14(q8)14{({p}^{4})}^{\frac{1}{4}}{({q}^{8})}^{\frac{1}{4}}
To raise a power to a power, we multiply the exponents.pq2p{q}^{2}

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

20. Example 9.118

Example 9.118

Example 9.118

Simplify: ⓐ x34⋅x−14x−64\frac{{x}^{\frac{3}{4}}\cdot{x}^{-\frac{1}{4}}}{{x}^{-\frac{6}{4}}} ⓑ y43⋅yy−23\frac{{y}^{\frac{4}{3}}\cdot{}y}{{y}^{-\frac{2}{3}}}.

Solution

ⓐ

x34⋅x−14x−64\frac{{x}^{\frac{3}{4}}\cdot{x}^{-\frac{1}{4}}}{{x}^{-\frac{6}{4}}}
Use the Product Property in the numerator, add the exponents.x24x−64\frac{{x}^{\frac{2}{4}}}{{x}^{-\frac{6}{4}}}
Use the Quotient Property, subtract the exponents.x84{x}^{\frac{8}{4}}
Simplify.x2{x}^{2}

ⓑ

y43⋅yy−23\frac{{y}^{\frac{4}{3}}\cdot{}y}{{y}^{-\frac{2}{3}}}
Use the Product Property in the numerator, add the exponents.y73y−23\frac{{y}^{\frac{7}{3}}}{{y}^{-\frac{2}{3}}}
Use the Quotient Property, subtract the exponents.y93{y}^{\frac{9}{3}}
Simplify.y3{y}^{3}

21. Section 9.8 Exercises

Section 9.8 Exercises

Section 9.8 Exercises

Simplify Expressions with a1n{a}^{\frac{1}{n}}

In the following exercises, write as a radical expression.

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Simplify Expressions with amn{a}^{\frac{m}{n}}

In the following exercises, write with a rational exponent.

In the following exercises, simplify.

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Everyday Math

Writing Exercises

22. Practice 1

  1. Write as a radical expression: ⓐ t12{t}^{\frac{1}{2}} ⓑ m13{m}^{\frac{1}{3}} ⓒ r14{r}^{\frac{1}{4}}.
  1. Write as a radial expression: ⓐ b12{b}^{\frac{1}{2}} ⓑ z13{z}^{\frac{1}{3}} ⓒ p14{p}^{\frac{1}{4}}.
  1. Write with a rational exponent: ⓐ s\sqrt{s} ⓑ x3\sqrt[3]{x} ⓒ b4\sqrt[4]{b}.
  1. Write with a rational exponent: ⓐ v\sqrt{v} ⓑ p3\sqrt[3]{p} ⓒ p4\sqrt[4]{p}.
  1. Write with a rational exponent: ⓐ 10m\sqrt{10m} ⓑ 3n5\sqrt[5]{3n} ⓒ 36y43\sqrt[4]{6y}.

23. Practice 2

  1. Write with a rational exponent: ⓐ 3k7\sqrt[7]{3k} ⓑ 5j4\sqrt[4]{5j} ⓒ 82a38\sqrt[3]{2a}.
  1. Simplify: ⓐ 3612{36}^{\frac{1}{2}} ⓑ 813{8}^{\frac{1}{3}} ⓒ 1614{16}^{\frac{1}{4}}.
  1. Simplify: ⓐ 10012{100}^{\frac{1}{2}} ⓑ 2713{27}^{\frac{1}{3}} ⓒ 8114{81}^{\frac{1}{4}}.
  1. Simplify: ⓐ (−125)13{(-125)}^{\frac{1}{3}} ⓑ −12513-{125}^{\frac{1}{3}} ⓒ (125)−13{(125)}^{-\frac{1}{3}}.
  1. Simplify: ⓐ (−32)15{(-32)}^{\frac{1}{5}} ⓑ −3215-{32}^{\frac{1}{5}} ⓒ (32)−15{(32)}^{-\frac{1}{5}}.

24. Practice 3

  1. Simplify: ⓐ (−64)12{(-64)}^{\frac{1}{2}} ⓑ −6412-{64}^{\frac{1}{2}} ⓒ (64)−12{(64)}^{-\frac{1}{2}}.
  1. Simplify: ⓐ (−256)14{(-256)}^{\frac{1}{4}} ⓑ −25614-{256}^{\frac{1}{4}} ⓒ (256)−14{(256)}^{-\frac{1}{4}}.
  1. Write with a rational exponent: ⓐ x5\sqrt{{x}^{5}} ⓑ z34\sqrt[4]{{z}^{3}} ⓒ y25\sqrt[5]{{y}^{2}}.
  1. Write with a rational exponent: ⓐ a25\sqrt[5]{{a}^{2}} ⓑ b73\sqrt[3]{{b}^{7}} ⓒ m54\sqrt[4]{{m}^{5}}.
  1. Simplify: ⓐ 432{4}^{\frac{3}{2}} ⓑ 2723{27}^{\frac{2}{3}} ⓒ 62534{625}^{\frac{3}{4}}.

25. Practice 4

  1. Simplify: ⓐ 853{8}^{\frac{5}{3}} ⓑ 8132{81}^{\frac{3}{2}} ⓒ 1634{16}^{\frac{3}{4}}.
  1. Simplify: ⓐ 8−53{8}^{-\frac{5}{3}} ⓑ 81−32{81}^{-\frac{3}{2}} ⓒ 16−34{16}^{-\frac{3}{4}}.
  1. Simplify: ⓐ 4−32{4}^{-\frac{3}{2}} ⓑ 27−23{27}^{-\frac{2}{3}} ⓒ 625−34{625}^{-\frac{3}{4}}.
  1. Simplify: ⓐ −1632{-16}^{\frac{3}{2}} ⓑ −16−32{-16}^{-\frac{3}{2}} ⓒ (−16)−32{(-16)}^{-\frac{3}{2}}.
  1. Simplify: ⓐ −8132{-81}^{\frac{3}{2}} ⓑ −81−32{-81}^{-\frac{3}{2}} ⓒ (−81)−32{(-81)}^{-\frac{3}{2}}.

26. Practice 5

  1. Simplify: ⓐ 323⋅343{3}^{\frac{2}{3}}\cdot{3}^{\frac{4}{3}} ⓑ y13⋅y83{y}^{\frac{1}{3}}\cdot{y}^{\frac{8}{3}} ⓒ m14⋅m34{m}^{\frac{1}{4}}\cdot{m}^{\frac{3}{4}}.
  1. Simplify: ⓐ 535⋅575{5}^{\frac{3}{5}}\cdot{5}^{\frac{7}{5}} ⓑ z18⋅z78{z}^{\frac{1}{8}}\cdot{z}^{\frac{7}{8}} ⓒ n27⋅n57{n}^{\frac{2}{7}}\cdot{n}^{\frac{5}{7}}.
  1. Simplify: ⓐ (p10)15{({p}^{10})}^{\frac{1}{5}} ⓑ (q8)34{({q}^{8})}^{\frac{3}{4}} ⓒ (x6)43{({x}^{6})}^{\frac{4}{3}}.
  1. Simplify: ⓐ (r6)53{({r}^{6})}^{\frac{5}{3}} ⓑ (s12)34{({s}^{12})}^{\frac{3}{4}} ⓒ (m9)29{({m}^{9})}^{\frac{2}{9}}.
  1. Simplify: ⓐ u54u14\frac{{u}^{\frac{5}{4}}}{{u}^{\frac{1}{4}}} ⓑ v35v25\frac{{v}^{\frac{3}{5}}}{{v}^{\frac{2}{5}}} ⓒ x23x53\frac{{x}^{\frac{2}{3}}}{{x}^{\frac{5}{3}}}.

27. Practice 6

  1. Simplify: ⓐ c125c25\frac{{c}^{\frac{12}{5}}}{{c}^{\frac{2}{5}}} ⓑ m54m94\frac{{m}^{\frac{5}{4}}}{{m}^{\frac{9}{4}}} ⓒ d15d65\frac{{d}^{\frac{1}{5}}}{{d}^{\frac{6}{5}}}.
  1. Simplify: ⓐ (32x13)35{(32{x}^{\frac{1}{3}})}^{\frac{3}{5}} ⓑ (64y23)13{(64{y}^{\frac{2}{3}})}^{\frac{1}{3}}.
  1. Simplify: ⓐ (16m13)32{(16{m}^{\frac{1}{3}})}^{\frac{3}{2}} ⓑ (81n25)32{(81{n}^{\frac{2}{5}})}^{\frac{3}{2}}.
  1. Simplify: ⓐ m23⋅m−13m−53\frac{{m}^{\frac{2}{3}}\cdot{m}^{-\frac{1}{3}}}{{m}^{-\frac{5}{3}}} ⓑ n16⋅nn−116\frac{{n}^{\frac{1}{6}}\cdot{}n}{{n}^{-\frac{11}{6}}}.
  1. Simplify: ⓐ u45⋅u−25u−135\frac{{u}^{\frac{4}{5}}\cdot{u}^{-\frac{2}{5}}}{{u}^{-\frac{13}{5}}} ⓑ v12⋅vv−72\frac{{v}^{\frac{1}{2}}\cdot{}v}{{v}^{-\frac{7}{2}}}.

28. Practice 7

  1. ⓐ r12{r}^{\frac{1}{2}} ⓑ s13{s}^{\frac{1}{3}} ⓒ t14{t}^{\frac{1}{4}}
  1. ⓐ g17{g}^{\frac{1}{7}} ⓑ h15{h}^{\frac{1}{5}} ⓒ j125{j}^{\frac{1}{25}}
  1. ⓐ r8\sqrt[8]{r} ⓑ

ⓒ t4\sqrt[4]{t}

  1. ⓐ u5\sqrt[5]{u} ⓑ v\sqrt{v} ⓒ
  1. ⓐ 5x4\sqrt[4]{5x} ⓑ 9y8\sqrt[8]{9y} ⓒ 73z57\sqrt[5]{3z}

29. Practice 8

  1. ⓐ 25a3\sqrt[3]{25a} ⓑ 3b\sqrt{3b} ⓒ
  1. ⓐ 62514{625}^{\frac{1}{4}} ⓑ 24315{243}^{\frac{1}{5}} ⓒ 3215{32}^{\frac{1}{5}}
  1. ⓐ 21613{216}^{\frac{1}{3}} ⓑ 3215{32}^{\frac{1}{5}} ⓒ 8114{81}^{\frac{1}{4}}
  1. ⓐ (−243)15{(-243)}^{\frac{1}{5}} ⓑ −24315-{243}^{\frac{1}{5}} ⓒ (243)−15{(243)}^{-\frac{1}{5}}
  1. ⓐ (−1000)13{(-1000)}^{\frac{1}{3}} ⓑ −100013-{1000}^{\frac{1}{3}} ⓒ (1000)−13{(1000)}^{-\frac{1}{3}}

30. Practice 9

  1. ⓐ (−49)12{(-49)}^{\frac{1}{2}} ⓑ −4912-{49}^{\frac{1}{2}} ⓒ (49)−12{(49)}^{-\frac{1}{2}}
  1. ⓐ (−1)14{(-1)}^{\frac{1}{4}} ⓑ (1)−14{(1)}^{-\frac{1}{4}} ⓒ −114-{1}^{\frac{1}{4}}
  1. ⓐ (−32)15{(-32)}^{\frac{1}{5}} ⓑ (243)−15{(243)}^{-\frac{1}{5}} ⓒ −12513-{125}^{\frac{1}{3}}
  1. ⓐ r74\sqrt[4]{{r}^{7}} ⓑ s35\sqrt[5]{{s}^{3}} ⓒ t73\sqrt[3]{{t}^{7}}
  1. ⓐ a3\sqrt[3]{a} ⓑ b5\sqrt{{b}^{5}} ⓒ c53\sqrt[3]{{c}^{5}}

31. Practice 10

  1. ⓐ 100023{1000}^{\frac{2}{3}} ⓑ 2532{25}^{\frac{3}{2}} ⓒ 3235{32}^{\frac{3}{5}}
  1. ⓐ 1632{16}^{\frac{3}{2}} ⓑ 12553{125}^{\frac{5}{3}} ⓒ 6443{64}^{\frac{4}{3}}
  1. ⓐ 6452{64}^{\frac{5}{2}} ⓑ 81−32{81}^{-\frac{3}{2}} ⓒ 27−43{27}^{-\frac{4}{3}}
  1. ⓐ 10032{100}^{\frac{3}{2}} ⓑ 49−52{49}^{-\frac{5}{2}} ⓒ (−100)32{(-100)}^{\frac{3}{2}}
  1. ⓐ −6432-{64}^{\frac{3}{2}} ⓑ −64−32-{64}^{-\frac{3}{2}} ⓒ (−64)32{(-64)}^{\frac{3}{2}}

32. Practice 11

  1. ⓐ −4932-{49}^{\frac{3}{2}} ⓑ −49−32-{49}^{-\frac{3}{2}} ⓒ (−49)32{(-49)}^{\frac{3}{2}}
  1. ⓐ 652⋅612{6}^{\frac{5}{2}}\cdot{6}^{\frac{1}{2}} ⓑ n210⋅n810{n}^{\frac{2}{10}}\cdot{n}^{\frac{8}{10}} ⓒ q25⋅q135{q}^{\frac{2}{5}}\cdot{q}^{\frac{13}{5}}
  1. ⓐ 1013⋅1053{10}^{\frac{1}{3}}\cdot{10}^{\frac{5}{3}} ⓑ x56⋅x76{x}^{\frac{5}{6}}\cdot{x}^{\frac{7}{6}} ⓒ y118⋅y218{y}^{\frac{11}{8}}\cdot{y}^{\frac{21}{8}}
  1. ⓐ (a12)16{({a}^{12})}^{\frac{1}{6}} ⓑ (b15)35{({b}^{15})}^{\frac{3}{5}} ⓒ (c11)111{({c}^{11})}^{\frac{1}{11}}
  1. ⓐ (h6)43{({h}^{6})}^{\frac{4}{3}} ⓑ (k12)34{({k}^{12})}^{\frac{3}{4}} ⓒ (j10)75{({j}^{10})}^{\frac{7}{5}}

33. Practice 12

  1. ⓐ s115s65\frac{{s}^{\frac{11}{5}}}{{s}^{\frac{6}{5}}} ⓑ z73z13\frac{{z}^{\frac{7}{3}}}{{z}^{\frac{1}{3}}} ⓒ w27w97\frac{{w}^{\frac{2}{7}}}{{w}^{\frac{9}{7}}}
  1. ⓐ u139u49\frac{{u}^{\frac{13}{9}}}{{u}^{\frac{4}{9}}} ⓑ r157r87\frac{{r}^{\frac{15}{7}}}{{r}^{\frac{8}{7}}} ⓒ n35n85\frac{{n}^{\frac{3}{5}}}{{n}^{\frac{8}{5}}}
  1. ⓐ (81r45)14{(81{r}^{\frac{4}{5}})}^{\frac{1}{4}} ⓑ (64s37)16{(64{s}^{\frac{3}{7}})}^{\frac{1}{6}}
  1. ⓐ (27m34)23{(27{m}^{\frac{3}{4}})}^{\frac{2}{3}} ⓑ (625n83)34{(625{n}^{\frac{8}{3}})}^{\frac{3}{4}}
  1. ⓐ (r8s4)14{({r}^{8}{s}^{4})}^{\frac{1}{4}} ⓑ (u15v20)15{({u}^{15}{v}^{20})}^{\frac{1}{5}}

34. Practice 13

  1. ⓐ (r16s10)12{({r}^{16}{s}^{10})}^{\frac{1}{2}} ⓑ (u10v5)45{({u}^{10}{v}^{5})}^{\frac{4}{5}}
  1. ⓐ a34⋅a−14a−104\frac{{a}^{\frac{3}{4}}\cdot{a}^{-\frac{1}{4}}}{{a}^{-\frac{10}{4}}} ⓑ b23⋅bb−73\frac{{b}^{\frac{2}{3}}\cdot{}b}{{b}^{-\frac{7}{3}}}
  1. ⓐ m74⋅m−54m−24\frac{{m}^{\frac{7}{4}}\cdot{m}^{-\frac{5}{4}}}{{m}^{-\frac{2}{4}}} ⓑ n37⋅nn−47\frac{{n}^{\frac{3}{7}}\cdot{}n}{{n}^{-\frac{4}{7}}}
  1. n26⋅n46{n}^{\frac{2}{6}}\cdot{n}^{\frac{4}{6}}
  1. (b10)35{({b}^{10})}^{\frac{3}{5}}

35. Practice 14

  1. z23z83\frac{{z}^{\frac{2}{3}}}{{z}^{\frac{8}{3}}}
  1. (64s35)16{(64{s}^{\frac{3}{5}})}^{\frac{1}{6}}
  1. (u12v18)16{({u}^{12}{v}^{18})}^{\frac{1}{6}}
  1. Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify 24212{242}^{\frac{1}{2}} to find the length of each side of his patio.Round to the nearest tenth of a foot.
  1. Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify 1024121612\frac{{1024}^{\frac{1}{2}}}{{16}^{\frac{1}{2}}} to find how many seconds it took for the flare to reach the water.

36. Practice 15

  1. Explain why the expression (−16)32{(-16)}^{\frac{3}{2}} cannot be evaluated.

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