9.8 Rational Exponents

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

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

\begin{matrix} {(8^{p})}^{3} & = & 8 \end{matrix}

Multiply the exponents on the left.

\begin{matrix} 8^{3 p} & = & 8 \end{matrix}

Write the exponent 1 on the right.

\begin{matrix} 8^{3 p} & = & 8^{1} \end{matrix}

The exponents must be equal.

3 p, equals, 1

Solve for

p .

\begin{matrix} p & = & \frac{1}{3} \end{matrix}

\begin{matrix} So {(8^{\frac{1}{3}})}^{3} & = & 8 . \end{matrix}

But we know also

{(\sqrt[3]{8})}^{3} = 8

. Then it must be that

8^{\frac{1}{3}} = \sqrt[3]{8}

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

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.

Example 1

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

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

ⓐ

	$x^{\frac{1}{2}}$
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.	$\sqrt{x}$

ⓑ

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

ⓒ

	$z^{\frac{1}{4}}$
The denominator of the exponent is 4, so the index is 4.	$the 4th root of z$

Example 2

Write with a rational exponent: ⓐ $\sqrt{x}$ ⓑ $\sqrt[3]{y}$ ⓒ $the 4th root of z$ .

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

ⓐ

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

ⓑ

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

ⓒ

	$the 4th root of z$
The index is 4, so the denominator of the exponent is 4.	$z^{\frac{1}{4}}$

Example 3

Write with a rational exponent: ⓐ $the square root of 5 y$ ⓑ $the cube root of 4 x$ ⓒ $3 the 4th root of 5 z$ .

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

ⓐ

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

ⓑ

	$the cube root of 4 x$
The index is 3, so the denominator of the exponent is 3.	${(4 x)}^{\frac{1}{3}}$

ⓒ

	$3 the 4th root of 5 z$
The index is 4, so the denominator of the exponent is 4.	$3 {(5 z)}^{\frac{1}{4}}$

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

Example 4

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

ⓐ

	$25^{\frac{1}{2}}$
Rewrite as a square root.	$the square root of 25$
Simplify.	$5$

ⓑ

	$64^{\frac{1}{3}}$
Rewrite as a cube root.	$the cube root of 64$
Recognize 64 is a perfect cube.	$the cube root of 4 cubed$
Simplify.	$4$

ⓒ

	$256^{\frac{1}{4}}$
Rewrite as a fourth root.	$the 4th root of 256$
Recognize 256 is a perfect fourth power.	$\sqrt[4]{4^{4}}$
Simplify.	$4$

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

Example 5

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

ⓐ

	${(−64)}^{\frac{1}{3}}$
Rewrite as a cube root.	$the cube root of −64$
Rewrite $−64$ as a perfect cube.	$the cube root of open parenthesis −4 close parenthesis cubed$
Simplify.	$−4$

ⓑ

	$− 64^{\frac{1}{3}}$
The exponent applies only to the 64.	$− (64^{\frac{1}{3}})$
Rewrite as a cube root.	$− the cube root of 64$
Rewrite 64 as $4 cubed.$	$− the cube root of 4 cubed$
Simplify.	$−4$

ⓒ

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

Example 6

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

ⓐ

	${(−16)}^{\frac{1}{4}}$
Rewrite as a fourth root.	$the 4th root of −16$
There is no real number whose fourth power is $−16.$

ⓑ

	$− 16^{\frac{1}{4}}$
The exponent only applies to the 16. Rewrite as a fourth root.	$− the 4th root of 16$
Rewrite $16$ as $2 to the power 4.$	$− \sqrt[4]{2^{4}}$
Simplify.	$−2$

ⓒ

	${(16)}^{- \frac{1}{4}}$
Rewrite using the property $a^{− n} = \frac{1}{a^{n}} .$	$\frac{1}{{(16)}^{\frac{1}{4}}}$
Rewrite as a fourth root.	$\frac{1}{\sqrt[4]{16}}$
Rewrite $16$ as $2 to the power 4.$	$\frac{1}{\sqrt[4]{2^{4}}}$
Simplify.	$\frac{1}{2}$

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

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

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

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

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

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

Example 8

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

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

ⓐ

	$9^{\frac{3}{2}}$
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.	$open parenthesis the square root of 9 close parenthesis cubed$
Simplify.	$open parenthesis 3 close parenthesis cubed$
	$27$

ⓑ

	$125^{\frac{2}{3}}$
The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.	$open parenthesis the cube root of 125 close parenthesis squared$
Simplify.	$open parenthesis 5 close parenthesis squared$
	$25$

ⓒ

	$81^{\frac{3}{4}}$
The power of the radical is the numerator of the exponent, 3. The index of the radical is the denominator of the exponent, 4.	$open parenthesis the 4th root of 81 close parenthesis cubed$
Simplify.	$open parenthesis 3 close parenthesis cubed$
	$27$

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

Example 9

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

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

ⓐ

	$16^{- \frac{3}{2}}$
Rewrite using $b^{− p} = \frac{1}{b^{p}} .$	$\frac{1}{16^{\frac{3}{2}}}$
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.	$\frac{1}{{(\sqrt{16})}^{3}}$
Simplify.	$\frac{1}{4^{3}}$
	$\frac{1}{64}$

ⓑ

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

ⓒ

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

Example 10

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

ⓐ

	$− 25^{\frac{3}{2}}$
Rewrite in radical form.	$− open parenthesis the square root of 25 close parenthesis cubed$
Simplify the radical.	$− open parenthesis 5 close parenthesis cubed$
Simplify.	$−125$

ⓑ

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

ⓒ

	${(−25)}^{\frac{3}{2}}$
Rewrite in radical form.	$open parenthesis the square root of −25 close parenthesis cubed$
There is no real number whose square root is $−25.$	Not a real number.

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.

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

Example 11

Simplify: ⓐ $2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}}$ ⓑ $x^{\frac{2}{3}} \cdot x^{\frac{4}{3}}$ ⓒ $z^{\frac{3}{4}} \cdot z^{\frac{5}{4}}$ .

ⓐ

	$2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}}$
The bases are the same, so we add the exponents.	$2^{\frac{1}{2} + \frac{5}{2}}$
Add the fractions.	$2^{\frac{6}{2}}$
Simplify the exponent.	$2 cubed$
Simplify.	$8$

ⓑ

	$x^{\frac{2}{3}} \cdot x^{\frac{4}{3}}$
The bases are the same, so we add the exponents.	$x^{\frac{2}{3} + \frac{4}{3}}$
Add the fractions.	$x^{\frac{6}{3}}$
Simplify.	$x^{2}$

ⓒ

	$z^{\frac{3}{4}} \cdot z^{\frac{5}{4}}$
The bases are the same, so we add the exponents.	$z^{\frac{3}{4} + \frac{5}{4}}$
Add the fractions.	$z^{\frac{8}{4}}$
Simplify.	$z^{2}$

We will use the Power Property in the next example.

Example 12

Simplify: ⓐ ${(x^{4})}^{\frac{1}{2}}$ ⓑ ${(y^{6})}^{\frac{1}{3}}$ ⓒ ${(z^{9})}^{\frac{2}{3}}$ .

ⓐ

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

ⓑ

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

ⓒ

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

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

Example 13

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

ⓐ

	$\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$
To divide with the same base, we subtract the exponents.	$x^{\frac{4}{3} - \frac{1}{3}}$
Simplify.	$x$

ⓑ

	$\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$
To divide with the same base, we subtract the exponents.	$y^{\frac{3}{4} - \frac{1}{4}}$
Simplify.	$y^{\frac{1}{2}}$

ⓒ

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

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.

Example 14

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

ⓐ

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

ⓑ

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

Example 15

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

ⓐ

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

ⓑ

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

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

Example 16

Simplify: ⓐ $\frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}}$ ⓑ $\frac{y^{\frac{4}{3}} \cdot y}{y^{- \frac{2}{3}}}$ .

ⓐ

	$\frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}}$
Use the Product Property in the numerator, add the exponents.	$\frac{x^{\frac{2}{4}}}{x^{- \frac{6}{4}}}$
Use the Quotient Property, subtract the exponents.	$x^{\frac{8}{4}}$
Simplify.	$x^{2}$

ⓑ

	$\frac{y^{\frac{4}{3}} \cdot y}{y^{- \frac{2}{3}}}$
Use the Product Property in the numerator, add the exponents.	$\frac{y^{\frac{7}{3}}}{y^{- \frac{2}{3}}}$
Use the Quotient Property, subtract the exponents.	$y^{\frac{9}{3}}$
Simplify.	$y^{3}$

Key Concepts

Summary of Exponent Properties
If a,b are real numbers and m,n are rational numbers, then
- Product Property $a^{m} \cdot a^{n} = a^{m + n}$
- Power Property ${(a^{m})}^{n} = a^{m \cdot n}$
- Product to a Power ${(a b)}^{m} = a^{m} b^{m}$
- Quotient Property:
  
  $\frac{a^{m}}{a^{n}} = a^{m - n}, a \neq 0, m > n$
  $\frac{a^{m}}{a^{n}} = \frac{1}{a^{n - m}}, a \neq 0, n > m$
- Zero Exponent Definition $a^{0} = 1$ , $a not equal to 0$
- Quotient to a Power Property ${(\frac{a}{b})}^{m} = \frac{a^{m}}{b^{m}}, b \neq 0$

Section Exercises

Practice Makes Perfect

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

In the following exercises, write as a radical expression.

ⓐ $x^{\frac{1}{2}}$ ⓑ $y^{\frac{1}{3}}$ ⓒ $z^{\frac{1}{4}}$

ⓐ $r^{\frac{1}{2}}$ ⓑ $s^{\frac{1}{3}}$ ⓒ $t^{\frac{1}{4}}$

ⓐ $\sqrt{r}$ ⓑ $\sqrt[3]{s}$ ⓒ $the 4th root of t$

ⓐ $u^{\frac{1}{5}}$ ⓑ $v^{\frac{1}{9}}$ ⓒ $w^{\frac{1}{20}}$

ⓐ $g^{\frac{1}{7}}$ ⓑ $h^{\frac{1}{5}}$ ⓒ $j^{\frac{1}{25}}$

ⓐ $the 7th root of g$ ⓑ $the 5th root of h$ ⓒ $the 25th root of j$

In the following exercises, write with a rational exponent.

ⓐ $- \sqrt[7]{x}$ ⓑ $the 9th root of y$ ⓒ $the 5th root of f$

ⓐ $the 8th root of r$ ⓑ

ⓒ

the 4th root of t

ⓐ $r^{\frac{1}{8}}$ ⓑ $s^{\frac{1}{10}}$ ⓒ $t^{\frac{1}{4}}$

ⓐ $\sqrt[3]{a}$ ⓑ

ⓒ

\sqrt{c}

ⓐ $the 5th root of u$ ⓑ $\sqrt{v}$ ⓒ

ⓐ $u^{\frac{1}{5}}$ ⓑ $v^{\frac{1}{2}}$ ⓒ $w^{\frac{1}{16}}$

ⓐ $the cube root of 7 c$ ⓑ $the 7th root of 12 d$ ⓒ $3 the 4th root of 5 f$

ⓐ $the 4th root of 5 x$ ⓑ $the 8th root of 9 y$ ⓒ $7 the 5th root of 3 z$

ⓐ ${(5 x)}^{\frac{1}{4}}$ ⓑ ${(9 y)}^{\frac{1}{8}}$ ⓒ $7 {(3 z)}^{\frac{1}{5}}$

ⓐ $the square root of 21 p$ ⓑ $the 4th root of 8 q$ ⓒ $4 the 6th root of 36 r$

ⓐ $the cube root of 25 a$ ⓑ $the square root of 3 b$ ⓒ

ⓐ ${(25 a)}^{\frac{1}{3}}$ ⓑ ${(3 b)}^{\frac{1}{2}}$ ⓒ ${(40 c)}^{\frac{1}{10}}$

In the following exercises, simplify.

ⓐ $81^{\frac{1}{2}}$ ⓑ $125^{\frac{1}{3}}$ ⓒ $64^{\frac{1}{2}}$

ⓐ $625^{\frac{1}{4}}$ ⓑ $243^{\frac{1}{5}}$ ⓒ $32^{\frac{1}{5}}$

ⓐ 5 ⓑ 3 ⓒ 2

ⓐ $16^{\frac{1}{4}}$ ⓑ $16^{\frac{1}{2}}$ ⓒ $3125^{\frac{1}{5}}$

ⓐ $216^{\frac{1}{3}}$ ⓑ $32^{\frac{1}{5}}$ ⓒ $81^{\frac{1}{4}}$

ⓐ 6 ⓑ 2 ⓒ 3

ⓐ ${(−216)}^{\frac{1}{3}}$ ⓑ $− 216^{\frac{1}{3}}$ ⓒ ${(216)}^{- \frac{1}{3}}$

ⓐ ${(−243)}^{\frac{1}{5}}$ ⓑ $− 243^{\frac{1}{5}}$ ⓒ ${(243)}^{- \frac{1}{5}}$

ⓐ $−3$ ⓑ $−3$ ⓒ $\frac{1}{3}$

ⓐ ${(−1)}^{\frac{1}{3}}$ ⓑ ${−1}^{\frac{1}{3}}$ ⓒ ${(1)}^{- \frac{1}{3}}$

ⓐ ${(−1000)}^{\frac{1}{3}}$ ⓑ $− 1000^{\frac{1}{3}}$ ⓒ ${(1000)}^{- \frac{1}{3}}$

ⓐ $−10$ ⓑ $−10$ ⓒ $\frac{1}{10}$

ⓐ ${(−81)}^{\frac{1}{4}}$ ⓑ $− 81^{\frac{1}{4}}$ ⓒ ${(81)}^{- \frac{1}{4}}$

ⓐ ${(−49)}^{\frac{1}{2}}$ ⓑ $− 49^{\frac{1}{2}}$ ⓒ ${(49)}^{- \frac{1}{2}}$

ⓐ not a real number ⓑ $−7$ ⓒ $\frac{1}{7}$

ⓐ ${(−36)}^{\frac{1}{2}}$ ⓑ $- 36^{\frac{1}{2}}$ ⓒ ${(36)}^{- \frac{1}{2}}$

ⓐ ${(−1)}^{\frac{1}{4}}$ ⓑ ${(1)}^{- \frac{1}{4}}$ ⓒ $− 1^{\frac{1}{4}}$

ⓐ not a real number ⓑ $1$ ⓒ $−1$

ⓐ ${(−100)}^{\frac{1}{2}}$ ⓑ $− 100^{\frac{1}{2}}$ ⓒ ${(100)}^{- \frac{1}{2}}$

ⓐ ${(−32)}^{\frac{1}{5}}$ ⓑ ${(243)}^{- \frac{1}{5}}$ ⓒ $− 125^{\frac{1}{3}}$

ⓐ $−2$ ⓑ $\frac{1}{3}$
ⓒ $−5$

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

In the following exercises, write with a rational exponent.

ⓐ $\sqrt{m^{5}}$ ⓑ $\sqrt[3]{n^{2}}$ ⓒ $the 4th root of p cubed$

ⓐ $\sqrt[4]{r^{7}}$ ⓑ $the 5th root of s cubed$ ⓒ $\sqrt[3]{t^{7}}$

ⓐ $r^{\frac{7}{4}}$ ⓑ $s^{\frac{3}{5}}$ ⓒ $t^{\frac{7}{3}}$

ⓐ $the 5th root of u squared$ ⓑ $\sqrt[5]{v^{8}}$ ⓒ $\sqrt[9]{w^{4}}$

ⓐ $\sqrt[3]{a}$ ⓑ $\sqrt{b^{5}}$ ⓒ $\sqrt[3]{c^{5}}$

ⓐ $a^{\frac{1}{3}}$ ⓑ $b^{\frac{5}{2}}$ ⓒ $c^{\frac{5}{3}}$

In the following exercises, simplify.

ⓐ $16^{\frac{3}{2}}$ ⓑ $8^{\frac{2}{3}}$ ⓒ ${10,000}^{\frac{3}{4}}$

ⓐ $1000^{\frac{2}{3}}$ ⓑ $25^{\frac{3}{2}}$ ⓒ $32^{\frac{3}{5}}$

ⓐ 100 ⓑ 125 ⓒ 8

ⓐ $27^{\frac{5}{3}}$ ⓑ $16^{\frac{5}{4}}$ ⓒ $32^{\frac{2}{5}}$

ⓐ $16^{\frac{3}{2}}$ ⓑ $125^{\frac{5}{3}}$ ⓒ $64^{\frac{4}{3}}$

ⓐ 64 ⓑ 3125 ⓒ 256

ⓐ $32^{\frac{2}{5}}$ ⓑ $27^{- \frac{2}{3}}$ ⓒ $25^{- \frac{3}{2}}$

ⓐ $64^{\frac{5}{2}}$ ⓑ $81^{- \frac{3}{2}}$ ⓒ $27^{- \frac{4}{3}}$

ⓐ 32,768 ⓑ $\frac{1}{729}$ ⓒ $\frac{1}{81}$

ⓐ $25^{\frac{3}{2}}$ ⓑ $9^{- \frac{3}{2}}$ ⓒ ${(−64)}^{\frac{2}{3}}$

ⓐ $100^{\frac{3}{2}}$ ⓑ $49^{- \frac{5}{2}}$ ⓒ ${(−100)}^{\frac{3}{2}}$

ⓐ 1000 ⓑ $\frac{1}{16,807}$ ⓒ not a real number

ⓐ $− 9^{\frac{3}{2}}$ ⓑ $− 9^{- \frac{3}{2}}$ ⓒ ${(−9)}^{\frac{3}{2}}$

ⓐ $− 64^{\frac{3}{2}}$ ⓑ $− 64^{- \frac{3}{2}}$ ⓒ ${(−64)}^{\frac{3}{2}}$

ⓐ $−512$
ⓑ $- \frac{1}{512}$ ⓒ not a real number

ⓐ $− 100^{\frac{3}{2}}$ ⓑ $− 100^{- \frac{3}{2}}$ ⓒ ${(−100)}^{\frac{3}{2}}$

ⓐ $− 49^{\frac{3}{2}}$ ⓑ $− 49^{- \frac{3}{2}}$ ⓒ ${(−49)}^{\frac{3}{2}}$

ⓐ $−343$ ⓑ $- \frac{1}{343}$ ⓒ not a real number

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

ⓐ $4^{\frac{5}{8}} \cdot 4^{\frac{11}{8}}$ ⓑ $m^{\frac{7}{12}} \cdot m^{\frac{17}{12}}$ ⓒ $p^{\frac{3}{7}} \cdot p^{\frac{18}{7}}$

ⓐ $6^{\frac{5}{2}} \cdot 6^{\frac{1}{2}}$ ⓑ $n^{\frac{2}{10}} \cdot n^{\frac{8}{10}}$ ⓒ $q^{\frac{2}{5}} \cdot q^{\frac{13}{5}}$

ⓐ 216 ⓑ $n$ ⓒ $q^{3}$

ⓐ $5^{\frac{1}{2}} \cdot 5^{\frac{7}{2}}$ ⓑ $c^{\frac{3}{4}} \cdot c^{\frac{9}{4}}$ ⓒ $d^{\frac{3}{5}} \cdot d^{\frac{2}{5}}$

ⓐ $10^{\frac{1}{3}} \cdot 10^{\frac{5}{3}}$ ⓑ $x^{\frac{5}{6}} \cdot x^{\frac{7}{6}}$ ⓒ $y^{\frac{11}{8}} \cdot y^{\frac{21}{8}}$

ⓐ 100 ⓑ $x^{2}$ ⓒ $y to the power 4$

ⓐ ${(m^{6})}^{\frac{5}{2}}$ ⓑ ${(n^{9})}^{\frac{4}{3}}$ ⓒ ${(p^{12})}^{\frac{3}{4}}$

ⓐ ${(a^{12})}^{\frac{1}{6}}$ ⓑ ${(b^{15})}^{\frac{3}{5}}$ ⓒ ${(c^{11})}^{\frac{1}{11}}$

ⓐ $a^{2}$ ⓑ $b to the power 9$
ⓒ $c$

ⓐ ${(x^{12})}^{\frac{2}{3}}$ ⓑ ${(y^{20})}^{\frac{2}{5}}$ ⓒ ${(z^{16})}^{\frac{1}{16}}$

ⓐ ${(h^{6})}^{\frac{4}{3}}$ ⓑ ${(k^{12})}^{\frac{3}{4}}$ ⓒ ${(j^{10})}^{\frac{7}{5}}$

ⓐ $h to the power 8$ ⓑ $k to the power 9$ ⓒ $j to the power 14$

ⓐ $\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}$ ⓑ $\frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}}$ ⓒ $\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}$

ⓐ $\frac{s^{\frac{11}{5}}}{s^{\frac{6}{5}}}$ ⓑ $\frac{z^{\frac{7}{3}}}{z^{\frac{1}{3}}}$ ⓒ $\frac{w^{\frac{2}{7}}}{w^{\frac{9}{7}}}$

ⓐ $s$ ⓑ $z^{2}$ ⓒ $\frac{1}{w}$

ⓐ $\frac{t^{\frac{12}{5}}}{t^{\frac{7}{5}}}$ ⓑ $\frac{x^{\frac{3}{2}}}{x^{\frac{1}{2}}}$ ⓒ $\frac{m^{\frac{13}{8}}}{m^{\frac{5}{8}}}$

ⓐ $\frac{u^{\frac{13}{9}}}{u^{\frac{4}{9}}}$ ⓑ $\frac{r^{\frac{15}{7}}}{r^{\frac{8}{7}}}$ ⓒ $\frac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}$

ⓐ $u$ ⓑ $r$ ⓒ $\frac{1}{n}$

ⓐ ${(9 p^{\frac{2}{3}})}^{\frac{5}{2}}$ ⓑ ${(27 q^{\frac{3}{2}})}^{\frac{4}{3}}$

ⓐ ${(81 r^{\frac{4}{5}})}^{\frac{1}{4}}$ ⓑ ${(64 s^{\frac{3}{7}})}^{\frac{1}{6}}$

ⓐ $3 r^{\frac{1}{5}}$ ⓑ $2 s^{\frac{1}{14}}$

ⓐ ${(16 u^{\frac{1}{3}})}^{\frac{3}{4}}$ ⓑ ${(100 v^{\frac{2}{5}})}^{\frac{3}{2}}$

ⓐ ${(27 m^{\frac{3}{4}})}^{\frac{2}{3}}$ ⓑ ${(625 n^{\frac{8}{3}})}^{\frac{3}{4}}$

ⓐ $9 m^{\frac{1}{2}}$ ⓑ $125 n squared$

ⓐ ${(x^{8} y^{10})}^{\frac{1}{2}}$ ⓑ ${(a^{9} b^{12})}^{\frac{1}{3}}$

ⓐ ${(r^{8} s^{4})}^{\frac{1}{4}}$ ⓑ ${(u^{15} v^{20})}^{\frac{1}{5}}$

ⓐ $r^{2} s$ ⓑ $u cubed v to the power 4$

ⓐ ${(a^{6} b^{16})}^{\frac{1}{2}}$ ⓑ ${(j^{9} k^{6})}^{\frac{2}{3}}$

ⓐ ${(r^{16} s^{10})}^{\frac{1}{2}}$ ⓑ ${(u^{10} v^{5})}^{\frac{4}{5}}$

ⓐ $r^{8} s^{5}$ ⓑ $u^{8} v^{4}$

ⓐ $\frac{r^{\frac{5}{2}} \cdot r^{- \frac{1}{2}}}{r^{- \frac{3}{2}}}$ ⓑ $\frac{s^{\frac{1}{5}} \cdot s}{s^{- \frac{9}{5}}}$

ⓐ $\frac{a^{\frac{3}{4}} \cdot a^{- \frac{1}{4}}}{a^{- \frac{10}{4}}}$ ⓑ $\frac{b^{\frac{2}{3}} \cdot b}{b^{- \frac{7}{3}}}$

ⓐ $a^{3}$ ⓑ $b to the power 4$

ⓐ $\frac{c^{\frac{5}{3}} \cdot c^{- \frac{1}{3}}}{c^{- \frac{2}{3}}}$ ⓑ $\frac{d^{\frac{3}{5}} \cdot d}{d^{- \frac{2}{5}}}$

ⓐ $\frac{m^{\frac{7}{4}} \cdot m^{- \frac{5}{4}}}{m^{- \frac{2}{4}}}$ ⓑ $\frac{n^{\frac{3}{7}} \cdot n}{n^{- \frac{4}{7}}}$

ⓐ $m$ ⓑ $n^{2}$

$4^{\frac{5}{2}} \cdot 4^{\frac{1}{2}}$

$n^{\frac{2}{6}} \cdot n^{\frac{4}{6}}$

$n$

${(a^{24})}^{\frac{1}{6}}$

${(b^{10})}^{\frac{3}{5}}$

$b to the power 6$

$\frac{w^{\frac{2}{5}}}{w^{\frac{7}{5}}}$

$\frac{z^{\frac{2}{3}}}{z^{\frac{8}{3}}}$

$\frac{1}{z^{2}}$

${(27 r^{\frac{3}{5}})}^{\frac{1}{3}}$

${(64 s^{\frac{3}{5}})}^{\frac{1}{6}}$

$2 s^{\frac{1}{10}}$

${(r^{9} s^{12})}^{\frac{1}{3}}$

${(u^{12} v^{18})}^{\frac{1}{6}}$

$u^{2} v^{3}$

Everyday Math

Landscaping Joe wants to have a square garden plot in his backyard. He has enough compost to cover an area of 144 square feet. Simplify $144^{\frac{1}{2}}$ to find the length of each side of his garden.

Landscaping Elliott wants to make a square patio in his yard. He has enough concrete to pave an area of 242 square feet. Simplify $242^{\frac{1}{2}}$ to find the length of each side of his patio.Round to the nearest tenth of a foot.

15.6 feet

Gravity While putting up holiday decorations, Bob dropped a decoration from the top of a tree that is 12 feet tall. Simplify $\frac{12^{\frac{1}{2}}}{16^{\frac{1}{2}}}$ to find how many seconds it took for the decoration to reach the ground. Round to the nearest tenth of a second.

Gravity An airplane dropped a flare from a height of 1024 feet above a lake. Simplify $\frac{1024^{\frac{1}{2}}}{16^{\frac{1}{2}}}$ to find how many seconds it took for the flare to reach the water.

8 seconds

Writing Exercises

Show two different algebraic methods to simplify $4^{\frac{3}{2}} .$ Explain all your steps.

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

Answers will vary.

Chapter 9 Review Exercises

Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

$the square root of 64$

$the square root of 144$

$- \sqrt{25}$

$- \sqrt{81}$

$−9$

$the square root of −9$

$the square root of −36$

not a real number

$\sqrt{64} + \sqrt{225}$

$\sqrt{64 + 225}$

Estimate Square Roots

In the following exercises, estimate each square root between two consecutive whole numbers.

$the square root of 28$

$the square root of 155$

$12 less than the square root of 155 less than 13$

Approximate Square Roots

In the following exercises, approximate each square root and round to two decimal places.

$the square root of 15$

$the square root of 57$

7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{q^{2}}$

$the square root of 64 b squared$

$8 vertical bar b vertical bar$

$− the square root of 121 a squared$

$the square root of 225 m squared n squared$

$15 vertical bar m vertical bar vertical bar n vertical bar$

$− the square root of 100 q squared$

$the square root of 49 y squared$

$7 vertical bar y vertical bar$

$the square root of 4 a squared b squared$

$the square root of 121 c squared d squared$

$11 vertical bar c vertical bar vertical bar d vertical bar$

Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$the square root of 300$

$the square root of 98$

$7 the square root of 2$

$\sqrt{x^{13}}$

$\sqrt{y^{19}}$

$| y^{9} | \sqrt{y}$

$\sqrt{16 m^{4}}$

$\sqrt{36 n^{13}}$

$6 n^{6} \sqrt{n}$

$\sqrt{288 m^{21}}$

$\sqrt{150 n^{7}}$

$5 vertical bar n cubed vertical bar the square root of 6 n$

$\sqrt{48 r^{5} s^{4}}$

$\sqrt{108 r^{5} s^{3}}$

$6 r squared vertical bar s vertical bar the square root of 3 r s$

$\frac{10 - \sqrt{50}}{5}$

$\frac{6 + \sqrt{72}}{6}$

$1 + \sqrt{2}$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{\frac{16}{25}}$

$\sqrt{\frac{81}{36}}$

$\frac{3}{2}$

$\sqrt{\frac{x^{8}}{x^{4}}}$

$\sqrt{\frac{y^{6}}{y^{2}}}$

$y^{2}$

$\sqrt{\frac{98 p^{6}}{2 p^{2}}}$

$\sqrt{\frac{72 q^{8}}{2 q^{4}}}$

$6 q squared$

$\sqrt{\frac{65}{121}}$

$\sqrt{\frac{26}{169}}$

$\frac{\sqrt{26}}{13}$

$\sqrt{\frac{64 x^{4}}{25 x^{2}}}$

$\sqrt{\frac{36 r^{10}}{16 r^{5}}}$

$\frac{3 r^{2} \sqrt{r}}{2}$

$\sqrt{\frac{48 p^{3} q^{5}}{27 p q}}$

$\sqrt{\frac{12 r^{5} s^{7}}{75 r^{2} s}}$

$\frac{2 | r | | s^{3} | \sqrt{r}}{5}$

Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

$3 \sqrt{2} + \sqrt{2}$

$5 \sqrt{5} + 7 \sqrt{5}$

$12 the square root of 5$

$4 \sqrt{y} + 4 \sqrt{y}$

$6 \sqrt{m} - 2 \sqrt{m}$

$4 the square root of m$

$−3 \sqrt{7} + 2 \sqrt{7} - \sqrt{7}$

$8 \sqrt{13} + 2 \sqrt{3} + 3 \sqrt{13}$

$11 \sqrt{13} + 2 \sqrt{3}$

$3 \sqrt{5 x y} - \sqrt{5 x y} + 3 \sqrt{5 x y}$

$2 \sqrt{3 r s} + \sqrt{3 r s} - 5 \sqrt{r s}$

$3 \sqrt{3 r s} - 5 \sqrt{r s}$

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

$\sqrt{32} + 3 \sqrt{2}$

$\sqrt{8} + 3 \sqrt{2}$

$5 the square root of 2$

$\sqrt{72} + \sqrt{50}$

$\sqrt{48} + \sqrt{75}$

$9 the square root of 3$

$3 \sqrt{32} + \sqrt{98}$

$\frac{1}{3} \sqrt{27} - \frac{1}{8} \sqrt{192}$

$0$

$\sqrt{50 y^{5}} - \sqrt{72 y^{5}}$

$6 \sqrt{18 n^{4}} - 3 \sqrt{8 n^{4}} + n^{2} \sqrt{50}$

$17 n squared the square root of 2$

Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

$\sqrt{2} \cdot \sqrt{20}$

$2 \sqrt{2} \cdot 6 \sqrt{14}$

$24 the square root of 7$

$\sqrt{2 m^{2}} \cdot \sqrt{20 m^{4}}$

$(6 \sqrt{2 y}) (3 \sqrt{50 y^{3}})$

$180 y squared$

$(6 \sqrt{3 v^{4}}) (5 \sqrt{30 v})$

$open parenthesis the square root of 8 close parenthesis squared$

$open parenthesis − the square root of 10 close parenthesis squared$

$(2 \sqrt{5}) (5 \sqrt{5})$

$(−3 \sqrt{3}) (5 \sqrt{18})$

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

$10 (2 - \sqrt{7})$

$20 - 10 \sqrt{7}$

$\sqrt{3} (4 + \sqrt{12})$

$(5 + \sqrt{2}) (3 - \sqrt{2})$

$13 - 2 \sqrt{2}$

$(5 - 3 \sqrt{7}) (1 - 2 \sqrt{7})$

$(1 - 3 \sqrt{x}) (5 + 2 \sqrt{x})$

$5 - 13 \sqrt{x} - 6 x$

$(3 + 4 \sqrt{y}) (10 - \sqrt{y})$

${(1 + 6 \sqrt{p})}^{2}$

$1 + 12 \sqrt{p} + 36 p$

${(2 - 6 \sqrt{5})}^{2}$

$(3 + 2 \sqrt{7}) (3 - 2 \sqrt{7})$

$−19$

$(6 - \sqrt{11}) (6 + \sqrt{11})$

Divide Square Roots

Divide Square Roots

In the following exercises, simplify.

$\frac{\sqrt{75}}{10}$

$\frac{\sqrt{3}}{2}$

$\frac{2 - \sqrt{12}}{6}$

$\frac{\sqrt{48}}{\sqrt{27}}$

$\frac{4}{3}$

$\frac{\sqrt{75 x^{7}}}{\sqrt{3 x^{3}}}$

$\frac{\sqrt{20 y^{5}}}{\sqrt{2 y}}$

$y squared the square root of 10$

$\frac{\sqrt{98 p^{6} q^{4}}}{\sqrt{2 p^{4} q^{8}}}$

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

$\frac{10}{\sqrt{15}}$

$\frac{2 \sqrt{15}}{3}$

$\frac{6}{\sqrt{6}}$

$\frac{5}{3 \sqrt{5}}$

$\frac{\sqrt{5}}{3}$

$\frac{10}{2 \sqrt{6}}$

$\sqrt{\frac{3}{28}}$

$\frac{\sqrt{21}}{14}$

$\sqrt{\frac{9}{75}}$

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

$\frac{4}{4 + \sqrt{27}}$

$\frac{16 - 12 \sqrt{3}}{−11}$

$\frac{5}{2 - \sqrt{10}}$

$\frac{4}{2 - \sqrt{5}}$

$−8 - 4 \sqrt{5}$

$\frac{5}{4 - \sqrt{8}}$

$\frac{\sqrt{2}}{\sqrt{p} + \sqrt{3}}$

$\frac{\sqrt{2 p} - \sqrt{6}}{p - 3}$

$\frac{\sqrt{x} - \sqrt{2}}{\sqrt{x} + \sqrt{2}}$

Solve Equations with Square Roots

Solve Radical Equations

In the following exercises, solve the equation.

$\sqrt{7 z + 1} = 6$

$\sqrt{4 u - 2} - 4 = 0$

$\sqrt{6 m + 4} - 5 = 0$

$\frac{7}{2}$

$\sqrt{2 u - 3} + 2 = 0$

$\sqrt{u - 4} + 4 = u$

4 and 5

$\sqrt{v - 9} + 9 = 0$

$\sqrt{r - 4} - r = −10$

$\sqrt{s - 9} - s = −9$

$2 \sqrt{2 x - 7} - 4 = 8$

$\frac{43}{2}$

$\sqrt{2 - x} = \sqrt{2 x - 7}$

$\sqrt{a} + 3 = \sqrt{a + 9}$

$\sqrt{r} + 3 = \sqrt{r + 4}$

$\sqrt{u} + 2 = \sqrt{u + 5}$

$\frac{1}{16}$

$\sqrt{n + 11} - 1 = \sqrt{n + 4}$

$\sqrt{y + 5} + 1 = \sqrt{2 y + 3}$

Use Square Roots in Applications

In the following exercises, solve. Round approximations to one decimal place.

A pallet of sod will cover an area of about 600 square feet. Trinh wants to order a pallet of sod to make a square lawn in his backyard. Use the formula $s = \sqrt{A}$ to find the length of each side of his lawn.

A helicopter dropped a package from a height of 900 feet above a stranded hiker. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the hiker.

7.5 seconds

Officer Morales measured the skid marks of one of the cars involved in an accident. The length of the skid marks was 245 feet. Use the formula $s = \sqrt{24 d}$ to find the speed of the car before the brakes were applied.

Higher Roots

Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $the 6th root of 64$
ⓑ $the cube root of 64$

ⓐ 2 ⓑ 4

ⓐ $the cube root of −27$
ⓑ $the 4th root of −64$

ⓐ $\sqrt[9]{d^{9}}$
ⓑ $\sqrt[8]{v^{8}}$

ⓐ $d$ ⓑ $| v |$

ⓐ $\sqrt[5]{a^{10}}$
ⓑ $\sqrt[3]{b^{27}}$

ⓐ $\sqrt[4]{16 x^{8}}$
ⓑ $\sqrt[6]{64 y^{12}}$

ⓐ $2 x squared$ ⓑ $2 y squared$

ⓐ $\sqrt[7]{128 r^{14}}$
ⓑ $\sqrt[4]{81 s^{24}}$

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[9]{d^{9}}$
ⓑ

ⓐ $d$ ⓑ $m \sqrt[11]{m^{6}}$

ⓐ $the cube root of 54$
ⓑ $the 4th root of 128$

ⓐ $\sqrt[5]{64 c^{8}}$
ⓑ $\sqrt[4]{48 d^{7}}$

ⓐ $2 c the 5th root of 2 c cubed$ ⓑ $2 vertical bar d vertical bar the 4th root of 3 d cubed$

ⓐ $\sqrt[3]{343 q^{7}}$
ⓑ $\sqrt[6]{192 r^{9}}$

ⓐ $the cube root of −500$
ⓑ $the 4th root of −16$

ⓐ $−5 the cube root of 4$ ⓑ not a real number

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

$\sqrt[5]{\frac{r^{10}}{r^{5}}}$

$\sqrt[3]{\frac{w^{12}}{w^{2}}}$

$w^{3} \sqrt[3]{w}$

$\sqrt[4]{\frac{64 y^{8}}{4 y^{5}}}$

$\sqrt[3]{\frac{54 z^{9}}{2 z^{3}}}$

$3 z squared$

$\sqrt[6]{\frac{64 a^{7}}{b^{2}}}$

Add and Subtract Higher Roots

In the following exercises, simplify.

$4 \sqrt[5]{20} - 2 \sqrt[5]{20}$

$2 the 5th root of 20$

$4 \sqrt[3]{18} + 3 \sqrt[3]{18}$

$\sqrt[4]{1250} - \sqrt[4]{162}$

$2 the 4th root of 2$

$\sqrt[3]{640 c^{5}} - \sqrt[3]{−80 c^{3}}$

$\sqrt[5]{96 t^{8}} + \sqrt[5]{486 t^{4}}$

$2 t \sqrt[5]{3 t^{3}} + 3 \sqrt[5]{2 t^{4}}$

Rational Exponents

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

In the following exercises, write as a radical expression.

$r^{\frac{1}{8}}$

$s^{\frac{1}{10}}$

In the following exercises, write with a rational exponent.

$the 5th root of u$

$the 6th root of v$

$v^{\frac{1}{6}}$

$the cube root of 9 m$

$the 6th root of 10 z$

${(10 z)}^{\frac{1}{6}}$

In the following exercises, simplify.

$16^{\frac{1}{4}}$

$32^{\frac{1}{5}}$

${(−125)}^{\frac{1}{3}}$

${(125)}^{- \frac{1}{3}}$

$\frac{1}{5}$

${(−9)}^{\frac{1}{2}}$

${(36)}^{- \frac{1}{2}}$

$\frac{1}{6}$

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

In the following exercises, write with a rational exponent.

$\sqrt[3]{q^{5}}$

$\sqrt[5]{n^{8}}$

$n^{\frac{8}{5}}$

In the following exercises, simplify.

$27^{- \frac{2}{3}}$

$64^{\frac{5}{2}}$

32,768

$36^{\frac{3}{2}}$

$81^{- \frac{5}{2}}$

$\frac{1}{59,049}$

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

$3^{\frac{4}{5}} \cdot 3^{\frac{6}{5}}$

${(x^{6})}^{\frac{4}{3}}$

$x to the power 8$

$\frac{z^{\frac{5}{2}}}{z^{\frac{7}{5}}}$

${(16 s^{\frac{9}{4}})}^{\frac{1}{4}}$

$2 s^{\frac{9}{16}}$

${(m^{8} n^{12})}^{\frac{1}{4}}$

$\frac{z^{\frac{2}{3}} \cdot z^{- \frac{1}{3}}}{z^{- \frac{5}{3}}}$

$z^{2}$

Practice Test

In the following exercises, simplify.

$\sqrt{81 + 144}$

$\sqrt{169 m^{4} n^{2}}$

$13 m squared vertical bar n vertical bar$

$\sqrt{36 n^{13}}$

$3 \sqrt{13} + 5 \sqrt{2} + \sqrt{13}$

$4 \sqrt{13} + 5 \sqrt{2}$

$5 \sqrt{20} + 2 \sqrt{125}$

$(3 \sqrt{6 y}) (2 \sqrt{50 y^{3}})$

$60 y squared the square root of 3$

$(2 - 5 \sqrt{x}) (3 + \sqrt{x})$

${(1 - 2 \sqrt{q})}^{2}$

$1 - 4 \sqrt{q} + 4 q$

ⓐ $\sqrt[4]{a^{12}}$
ⓑ $\sqrt[3]{b^{21}}$

ⓐ $\sqrt[4]{81 x^{12}}$
ⓑ $\sqrt[6]{64 y^{18}}$

ⓐ $3 vertical bar x cubed vertical bar$ ⓑ $2 vertical bar y cubed vertical bar$

$\sqrt{\frac{64 r^{12}}{25 r^{6}}}$

$\sqrt{\frac{14 y^{3}}{7 y}}$

$vertical bar y vertical bar the square root of 2$

$\frac{\sqrt[5]{256 x^{7}}}{\sqrt[5]{4 x^{2}}}$

$\sqrt[4]{512} - 2 \sqrt[4]{32}$

$0$

ⓐ $256^{\frac{1}{4}}$
ⓑ $243^{\frac{1}{5}}$

$49^{\frac{3}{2}}$

343

$25^{- \frac{5}{2}}$

$\frac{w^{\frac{3}{4}}}{w^{\frac{7}{4}}}$

$\frac{1}{w}$

${(27 s^{\frac{3}{5}})}^{\frac{1}{3}}$

In the following exercises, rationalize the denominator.

$\frac{3}{2 \sqrt{6}}$

$\frac{\sqrt{6}}{4}$

$\frac{\sqrt{3}}{\sqrt{x} + \sqrt{5}}$

In the following exercises, solve.

$3 \sqrt{2 x - 3} - 20 = 7$

$\sqrt{3 u - 2} = \sqrt{5 u + 1}$

In the following exercise, solve.

A helicopter flying at an altitude of 600 feet dropped a package to a lifeboat. Use the formula $t = \frac{\sqrt{h}}{4}$ to find how many seconds it took for the package to reach the lifeboat. Round your answer to the nearest tenth of a second.

6.1 seconds

Key terms

rational exponents

If $\sqrt[n]{a}$ is a real number and $n greater than or equal to 2$ , $a^{\frac{1}{n}} = \sqrt[n]{a}$ .
For any positive integers m and n, $a^{\frac{m}{n}} = {(\sqrt[n]{a})}^{m}$ and $a^{\frac{m}{n}} = \sqrt[n]{a^{m}}$ .

Adapted from Elementary Algebra 2e by OpenStax (openstax.org), licensed under CC BY-NC-SA 4.0. Changes were made. License: CC-BY-NC-SA-4.0.

9.8 Rational Exponents

Simplify Expressions with a1n

Simplify Expressions with amn

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

Key Concepts

Section Exercises

Practice Makes Perfect

Everyday Math

Writing Exercises

Chapter 9 Review Exercises

Simplify and Use Square Roots

Simplify Square Roots

Add and Subtract Square Roots

Multiply Square Roots

Divide Square Roots

Solve Equations with Square Roots

Higher Roots

Rational Exponents

Practice Test

Key terms

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

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