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.

Step-by-step solution for an exponential equation, illustrating the relationship between fractional exponents and roots.
	$\begin{matrix} {(8^{p})}^{3} & = & 8 \end{matrix}$
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.	$\begin{matrix} 3 p & = & 1 \end{matrix}$
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 $8^{\frac{1}{3}} = \sqrt[3]{8}$ .

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.

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

Solution

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

ⓐ

This table illustrates the conversion of a fractional exponent (x^(1/2)) to its equivalent radical form (sqrt(x)), including a rule explanation.
	$x^{\frac{1}{2}}$
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.	$\sqrt{x}$

ⓑ

This table demonstrates the relationship between fractional exponents and their radical equivalents, showing how the exponent's denominator determines the radical's index.
	$y^{\frac{1}{3}}$
The denominator of the exponent is 3, so the index is 3.	$\sqrt[3]{y}$

ⓒ

Conversion of fractional exponents to radical form, showing the exponent's denominator as the radical's index.
	$z^{\frac{1}{4}}$
The denominator of the exponent is 4, so the index is 4.	$\sqrt[4]{z}$

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

Solution

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

ⓐ

Illustrates the equivalence between square root notation and fractional exponent notation, with explanations for the transformation.
	$\sqrt{x}$
No index is shown, so it is 2. The denominator of the exponent will be 2.	$x^{\frac{1}{2}}$

ⓑ

Conversion of cube roots to fractional exponents, illustrating how the root's index determines the exponent's denominator.
	$\sqrt[3]{y}$
The index is 3, so the denominator of the exponent is 3.	$y^{\frac{1}{3}}$

ⓒ

Illustrates the conversion of a fourth root expression to its equivalent exponential form, detailing the rule for the exponent's denominator.
	$\sqrt[4]{z}$
The index is 4, so the denominator of the exponent is 4.	$z^{\frac{1}{4}}$

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

Solution

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

ⓐ

Illustrates converting radical expressions to rational exponent form, showing how the root's index determines the fractional exponent.
	$\sqrt{5 y}$
No index is shown, so it is 2. The denominator of the exponent will be 2.	${(5 y)}^{\frac{1}{2}}$

ⓑ

Illustrates converting the cube root of 4x to its equivalent exponential form with a fractional exponent, explaining the role of the index.
	$\sqrt[3]{4 x}$
The index is 3, so the denominator of the exponent is 3.	${(4 x)}^{\frac{1}{3}}$

ⓒ

Conversion of a radical expression to an equivalent exponential form, showing an example with an explanation.
	$3 \sqrt[4]{5 z}$
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.

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

Solution

ⓐ

This table illustrates the step-by-step simplification of a numerical expression with a fractional exponent.
	$25^{\frac{1}{2}}$
Rewrite as a square root.	$\sqrt{25}$
Simplify.	$5$

ⓑ

This table illustrates the step-by-step simplification of a numerical expression with a fractional exponent, converting it to a cube root for evaluation.
	$64^{\frac{1}{3}}$
Rewrite as a cube root.	$\sqrt[3]{64}$
Recognize 64 is a perfect cube.	$\sqrt[3]{4^{3}}$
Simplify.	$4$

ⓒ

Step-by-step simplification of 256^(1/4), demonstrating rewriting as a root and final simplification.
	$256^{\frac{1}{4}}$
Rewrite as a fourth root.	$\sqrt[4]{256}$
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.

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

Solution

ⓐ

Illustrates the step-by-step evaluation of a fractional exponent expression by conversion to a cube root and simplification.
	${(−64)}^{\frac{1}{3}}$
Rewrite as a cube root.	$\sqrt[3]{−64}$
Rewrite $−64$ as a perfect cube.	$\sqrt[3]{{(−4)}^{3}}$
Simplify.	$−4$

ⓑ

Detailed steps for simplifying the mathematical expression -64^(1/3) by rewriting it as a cube root and evaluating the final result.
	$− 64^{\frac{1}{3}}$
The exponent applies only to the 64.	$− (64^{\frac{1}{3}})$
Rewrite as a cube root.	$− \sqrt[3]{64}$
Rewrite 64 as $4^{3} .$	$− \sqrt[3]{4^{3}}$
Simplify.	$−4$

ⓒ

This table demonstrates the step-by-step process for simplifying a mathematical expression with a negative fractional exponent.
	${(64)}^{- \frac{1}{3}}$
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 $4^{3} .$	$\frac{1}{\sqrt[3]{4^{3}}}$
Simplify.	$\frac{1}{4}$

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

Solution

ⓐ

Evaluation of (-16)^(1/4), illustrating it has no real number solution.
	${(−16)}^{\frac{1}{4}}$
Rewrite as a fourth root.	$\sqrt[4]{−16}$
There is no real number whose fourth power is $−16 .$

ⓑ

Step-by-step simplification of the mathematical expression -16^(1/4) to its final integer value.
	$− 16^{\frac{1}{4}}$
The exponent only applies to the 16. Rewrite as a fourth root.	$− \sqrt[4]{16}$
Rewrite $16$ as $2^{4} .$	$− \sqrt[4]{2^{4}}$
Simplify.	$−2$

ⓒ

This table shows the step-by-step simplification of the mathematical expression (16)^(-1/4), transforming it into its simplified fractional form of 1/2.
	${(16)}^{- \frac{1}{4}}$
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 $2^{4} .$	$\frac{1}{\sqrt[4]{2^{4}}}$
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.

Steps to simplify an exponential expression with a fractional power, demonstrating its equivalence to a radical form.
	${(a^{\frac{1}{n}})}^{m}$
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.

This table demonstrates the steps to simplify (a^m)^(1/n) to a^(m/n), illustrating its equivalence with the n-th root of (a^m).
	${(a^{m})}^{\frac{1}{n}}$
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.

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

Solution

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

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

Solution

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.

ⓐ

This table illustrates the step-by-step evaluation of the exponential expression 9^(3/2) by converting it to radical 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.	${(\sqrt{9})}^{3}$
Simplify.	${(3)}^{3}$
	$27$

ⓑ

Steps to simplify an expression with a rational exponent by converting it to radical form.
	$125^{\frac{2}{3}}$
The power of the radical is the numerator of the exponent, 2. The index of the radical is the denominator of the exponent, 3.	${(\sqrt[3]{125})}^{2}$
Simplify.	${(5)}^{2}$
	$25$

ⓒ

Steps to simplify an expression with a rational exponent by converting it to radical form and calculating the result.
	$81^{\frac{3}{4}}$
The power of the radical is the numerator of the exponent, 3. The index of the radical is the denominator of the exponent, 4.	${(\sqrt[4]{81})}^{3}$
Simplify.	${(3)}^{3}$
	$27$

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

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

Solution

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

ⓐ

Step-by-step simplification of an expression with a negative fractional exponent, showing the application of exponent and radical rules.
	$16^{- \frac{3}{2}}$
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.	$\frac{1}{4^{3}}$
	$\frac{1}{64}$

ⓑ

Detailed steps to simplify 32^(-2/5) by converting to a fraction, radical form, and evaluating to 1/4.
	$32^{- \frac{2}{5}}$
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.	$\frac{1}{2^{2}}$
	$\frac{1}{4}$

ⓒ

Step-by-step simplification of an expression with a negative fractional exponent, demonstrating conversions to positive exponents and radical form.
	$4^{- \frac{5}{2}}$
Rewrite using $b^{− p} = \frac{1}{b^{p}} .$	$\frac{1}{4^{\frac{5}{2}}}$
Change to radical form.	$\frac{1}{{(\sqrt{4})}^{5}}$
Simplify.	$\frac{1}{2^{5}}$
	$\frac{1}{32}$

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

Solution

ⓐ

Step-by-step simplification of the expression -25^(3/2) by rewriting it in radical form and evaluating it to -125.
	$− 25^{\frac{3}{2}}$
Rewrite in radical form.	$− {(\sqrt{25})}^{3}$
Simplify the radical.	$− {(5)}^{3}$
Simplify.	$−125$

ⓑ

This table illustrates the step-by-step simplification of the mathematical expression -25^(-3/2) by applying exponent rules and converting to radical form.
	$− 25^{- \frac{3}{2}}$
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.	$- \frac{1}{125}$

ⓒ

This table evaluates the expression (-25)^(3/2), demonstrating its transformation into radical form and concluding it is not a real number.
	${(−25)}^{\frac{3}{2}}$
Rewrite in radical form.	${(\sqrt{−25})}^{3}$
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.

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

Solution

ⓐ

Step-by-step simplification of the exponential expression 2^(1/2) * 2^(5/2) to 8.
	$2^{\frac{1}{2}} \cdot 2^{\frac{5}{2}}$
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.	$2^{3}$
Simplify.	$8$

ⓑ

Steps demonstrating the simplification of an algebraic expression involving the multiplication of exponential terms with the same base.
	$x^{\frac{2}{3}} \cdot x^{\frac{4}{3}}$
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.	$x^{2}$

ⓒ

Step-by-step simplification of an algebraic expression with fractional exponents using exponent rules.
	$z^{\frac{3}{4}} \cdot z^{\frac{5}{4}}$
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.

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

Solution

ⓐ

Simplifying an expression with a power raised to a power by multiplying exponents, with steps and examples.
	${(x^{4})}^{\frac{1}{2}}$
To raise a power to a power, we multiply the exponents.	$x^{4 \cdot \frac{1}{2}}$
Simplify.	$x^{2}$

ⓑ

This table illustrates the step-by-step simplification of an algebraic expression involving a power raised to a fractional power, applying the rule for multiplying exponents.
	${(y^{6})}^{\frac{1}{3}}$
To raise a power to a power, we multiply the exponents.	$y^{6 \cdot \frac{1}{3}}$
Simplify.	$y^{2}$

ⓒ

Demonstration of simplifying the exponential expression (z^9)^(2/3) by applying the power to a power rule.
	${(z^{9})}^{\frac{2}{3}}$
To raise a power to a power, we multiply the exponents.	$z^{9 \cdot \frac{2}{3}}$
Simplify.	$z^{6}$

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

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

Solution

ⓐ

Step-by-step simplification of an exponential expression involving division with the same base by subtracting fractional exponents.
	$\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$
To divide with the same base, we subtract the exponents.	$x^{\frac{4}{3} - \frac{1}{3}}$
Simplify.	$x$

ⓑ

Steps to simplify an exponential expression by dividing terms with the same base and subtracting their fractional exponents.
	$\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$
To divide with the same base, we subtract the exponents.	$y^{\frac{3}{4} - \frac{1}{4}}$
Simplify.	$y^{\frac{1}{2}}$

ⓒ

This table demonstrates the simplification of an exponential expression by applying rules for dividing with the same base and rewriting negative exponents.
	$\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}$
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.

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

Solution

ⓐ

Demonstrates the step-by-step simplification of the expression (27u^(1/2))^(2/3) to 9u^(1/3) using exponent rules.
	${(27 u^{\frac{1}{2}})}^{\frac{2}{3}}$
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.	$9 u^{\frac{1}{3}}$

ⓑ

This table demonstrates the step-by-step simplification of an algebraic expression involving rational exponents.
	${(8 v^{\frac{1}{4}})}^{\frac{2}{3}}$
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.	$4 v^{\frac{1}{6}}$

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

Solution

ⓐ

Step-by-step simplification of a mathematical expression involving powers and roots using the Product to a Power Property.
	${(m^{3} n^{9})}^{\frac{1}{3}}$
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.	$m n^{3}$

ⓑ

Step-by-step example illustrating the simplification of an exponential expression (p^4q^8)^(1/4) using exponent properties.
	${(p^{4} q^{8})}^{\frac{1}{4}}$
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.

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

Solution

ⓐ

This table illustrates the step-by-step simplification of an algebraic expression involving fractional exponents using properties of exponents.
	$\frac{x^{\frac{3}{4}} \cdot x^{- \frac{1}{4}}}{x^{- \frac{6}{4}}}$
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.	$x^{2}$

ⓑ

Step-by-step simplification of an algebraic expression using product and quotient properties of exponents.
	$\frac{y^{\frac{4}{3}} \cdot y}{y^{- \frac{2}{3}}}$
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 $a^{0} = 1$ , $a \neq 0$
- 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.

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

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

Solution

ⓐ $\sqrt{r}$ ⓑ $\sqrt[3]{s}$ ⓒ $\sqrt[4]{t}$

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

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

Solution

ⓐ $\sqrt[7]{g}$ ⓑ $\sqrt[5]{h}$ ⓒ $\sqrt[25]{j}$

In the following exercises, write with a rational exponent.

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

ⓐ $\sqrt[8]{r}$ ⓑ A mathematical radical symbol is displayed, with '19' as the index above the left arm of the root and '5' as the radicand inside, indicating the 19th root of 5. ⓒ $\sqrt[4]{t}$

Solution

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

ⓐ $\sqrt[3]{a}$ ⓑ The mathematical expression for the twelfth root of 'b'. ⓒ $\sqrt{c}$

ⓐ $\sqrt[5]{u}$ ⓑ $\sqrt{v}$ ⓒ A close-up view of the mathematical expression '16th root of W', stylized as a radical sign with '16' as the index and 'W' as the radicand.

Solution

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

ⓐ $\sqrt[3]{7 c}$ ⓑ $\sqrt[7]{12 d}$ ⓒ $3 \sqrt[4]{5 f}$

ⓐ $\sqrt[4]{5 x}$ ⓑ $\sqrt[8]{9 y}$ ⓒ $7 \sqrt[5]{3 z}$

Solution

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

ⓐ $\sqrt{21 p}$ ⓑ $\sqrt[4]{8 q}$ ⓒ $4 \sqrt[6]{36 r}$

ⓐ $\sqrt[3]{25 a}$ ⓑ $\sqrt{3 b}$ ⓒ A mathematical expression showing the tenth root of 40c, represented as an n-th root symbol with 10 as the index and 40c as the radicand.

Solution

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

In the following exercises, simplify.

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

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

Solution

ⓐ 5 ⓑ 3 ⓒ 2

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

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

Solution

ⓐ 6 ⓑ 2 ⓒ 3

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

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

Solution

ⓐ $−3$ ⓑ $−3$ ⓒ $\frac{1}{3}$

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

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

Solution

ⓐ $−10$ ⓑ $−10$ ⓒ $\frac{1}{10}$

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

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

Solution

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

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

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

Solution

ⓐ not a real number ⓑ $1$ ⓒ $−1$

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

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

Solution

ⓐ $−2$ ⓑ $\frac{1}{3}$
ⓒ $−5$

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

In the following exercises, write with a rational exponent.

ⓐ $\sqrt{m^{5}}$ ⓑ $\sqrt[3]{n^{2}}$ ⓒ $\sqrt[4]{p^{3}}$

ⓐ $\sqrt[4]{r^{7}}$ ⓑ $\sqrt[5]{s^{3}}$ ⓒ $\sqrt[3]{t^{7}}$

Solution

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

ⓐ $\sqrt[5]{u^{2}}$ ⓑ $\sqrt[5]{v^{8}}$ ⓒ $\sqrt[9]{w^{4}}$

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

Solution

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

In the following exercises, simplify.

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

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

Solution

ⓐ 100 ⓑ 125 ⓒ 8

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

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

Solution

ⓐ 64 ⓑ 3125 ⓒ 256

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

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

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

Solution

ⓐ 216 ⓑ $n$ ⓒ $q^{3}$

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

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

Solution

ⓐ 100 ⓑ $x^{2}$ ⓒ $y^{4}$

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

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

Solution

ⓐ $a^{2}$ ⓑ $b^{9}$
ⓒ $c$

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

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

Solution

ⓐ $h^{8}$ ⓑ $k^{9}$ ⓒ $j^{14}$

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

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

Solution

ⓐ $s$ ⓑ $z^{2}$ ⓒ $\frac{1}{w}$

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

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

Solution

ⓐ $u$ ⓑ $r$ ⓒ $\frac{1}{n}$

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

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

Solution

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

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

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

Solution

ⓐ $9 m^{\frac{1}{2}}$ ⓑ $125 n^{2}$

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

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

Solution

ⓐ $r^{2} s$ ⓑ $u^{3} v^{4}$

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

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

Solution

ⓐ $r^{8} s^{5}$ ⓑ $u^{8} v^{4}$

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

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

Solution

ⓐ $a^{3}$ ⓑ $b^{4}$

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

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

Solution

ⓐ $m$ ⓑ $n^{2}$

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

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

Solution

$n$

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

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

Solution

$b^{6}$

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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.

Solution

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.

Solution

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.

Solution

Answers will vary.

Chapter 9 Review Exercises

Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{64}$

$\sqrt{144}$

Solution

$- \sqrt{25}$

$- \sqrt{81}$

Solution

$−9$

$\sqrt{−9}$

$\sqrt{−36}$

Solution

not a real number

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

$\sqrt{64 + 225}$

Solution

Estimate Square Roots

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

$\sqrt{28}$

$\sqrt{155}$

Solution

$12 < \sqrt{155} < 13$

Approximate Square Roots

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

$\sqrt{15}$

$\sqrt{57}$

Solution

7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

$\sqrt{q^{2}}$

$\sqrt{64 b^{2}}$

Solution

$8 |b|$

$− \sqrt{121 a^{2}}$

$\sqrt{225 m^{2} n^{2}}$

Solution

$15 |m| |n|$

$− \sqrt{100 q^{2}}$

$\sqrt{49 y^{2}}$

Solution

$7 |y|$

$\sqrt{4 a^{2} b^{2}}$

$\sqrt{121 c^{2} d^{2}}$

Solution

$11 |c| |d|$

Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

$\sqrt{300}$

$\sqrt{98}$

Solution

$7 \sqrt{2}$

$\sqrt{x^{13}}$

$\sqrt{y^{19}}$

Solution

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

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

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

Solution

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

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

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

Solution

$5 |n^{3}| \sqrt{6 n}$

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

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

Solution

$6 r^{2} |s| \sqrt{3 r s}$

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

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

Solution

$1 + \sqrt{2}$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

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

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

Solution

$\frac{3}{2}$

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

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

Solution

$y^{2}$

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

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

Solution

$6 q^{2}$

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

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

Solution

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

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

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

Solution

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

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

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

Solution

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

Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

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

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

Solution

$12 \sqrt{5}$

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

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

Solution

$4 \sqrt{m}$

$−3 \sqrt{7} + 2 \sqrt{7} - \sqrt{7}$

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

Solution

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

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

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

Solution

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

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

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

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

Solution

$5 \sqrt{2}$

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

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

Solution

$9 \sqrt{3}$

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

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

Solution

$0$

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

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

Solution

$17 n^{2} \sqrt{2}$

Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

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

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

Solution

$24 \sqrt{7}$

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

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

Solution

$180 y^{2}$

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

${(\sqrt{8})}^{2}$

Solution

${(− \sqrt{10})}^{2}$

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

Solution

$(−3 \sqrt{3}) (5 \sqrt{18})$

Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

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

Solution

$20 - 10 \sqrt{7}$

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

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

Solution

$13 - 2 \sqrt{2}$

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

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

Solution

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

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

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

Solution

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

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

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

Solution

$−19$

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

Divide Square Roots

Divide Square Roots

In the following exercises, simplify.

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

Solution

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

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

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

Solution

$\frac{4}{3}$

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

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

Solution

$y^{2} \sqrt{10}$

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

Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

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

Solution

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

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

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

Solution

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

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

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

Solution

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

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

Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

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

Solution

$\frac{16 - 12 \sqrt{3}}{−11}$

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

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

Solution

$−8 - 4 \sqrt{5}$

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

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

Solution

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

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

Solve Equations with Square Roots

Solve Radical Equations

In the following exercises, solve the equation.

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

Solution

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

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

Solution

$\frac{7}{2}$

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

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

Solution

4 and 5

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

$\sqrt{r - 4} - r = −10$

Solution

$\sqrt{s - 9} - s = −9$

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

Solution

$\frac{43}{2}$

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

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

Solution

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

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

Solution

$\frac{1}{16}$

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

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

Solution

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.

Solution

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.

ⓐ $\sqrt[6]{64}$
ⓑ $\sqrt[3]{64}$

Solution

ⓐ 2 ⓑ 4

ⓐ $\sqrt[3]{−27}$
ⓑ $\sqrt[4]{−64}$

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

Solution

ⓐ $d$ ⓑ $| v |$

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

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

Solution

ⓐ $2 x^{2}$ ⓑ $2 y^{2}$

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

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

ⓐ $\sqrt[9]{d^{9}}$
ⓑ A mathematical expression showing the 11th root of m raised to the power of 17, written as '11th root of m^17'.

Solution

ⓐ $d$ ⓑ $m \sqrt[11]{m^{6}}$

ⓐ $\sqrt[3]{54}$
ⓑ $\sqrt[4]{128}$

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

Solution

ⓐ $2 c \sqrt[5]{2 c^{3}}$ ⓑ $2 |d| \sqrt[4]{3 d^{3}}$

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

ⓐ $\sqrt[3]{−500}$
ⓑ $\sqrt[4]{−16}$

Solution

ⓐ $−5 \sqrt[3]{4}$ ⓑ not a real number

Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

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

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

Solution

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

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

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

Solution

$3 z^{2}$

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

Add and Subtract Higher Roots

In the following exercises, simplify.

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

Solution

$2 \sqrt[5]{20}$

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

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

Solution

$2 \sqrt[4]{2}$

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

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

Solution

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

Rational Exponents

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

In the following exercises, write as a radical expression.

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

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

Solution

A close-up image showing a mathematical expression: the square root of nineteen divided by S. It appears to be a variable or a value S under the radical symbol, with 19 also inside.

In the following exercises, write with a rational exponent.

$\sqrt[5]{u}$

$\sqrt[6]{v}$

Solution

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

$\sqrt[3]{9 m}$

$\sqrt[6]{10 z}$

Solution

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

In the following exercises, simplify.

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

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

Solution

${(−125)}^{\frac{1}{3}}$

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

Solution

$\frac{1}{5}$

${(−9)}^{\frac{1}{2}}$

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

Solution

$\frac{1}{6}$

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

In the following exercises, write with a rational exponent.

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

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

Solution

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

In the following exercises, simplify.

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

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

Solution

32,768

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

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

Solution

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

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

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

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

Solution

$x^{8}$

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

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

Solution

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

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

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

Solution

$z^{2}$

Practice Test

In the following exercises, simplify.

$\sqrt{81 + 144}$

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

Solution

$13 m^{2} | n |$

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

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

Solution

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

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

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

Solution

$60 y^{2} \sqrt{3}$

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

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

Solution

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

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

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

Solution

ⓐ $3 |x^{3}|$ ⓑ $2 |y^{3}|$

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

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

Solution

$|y| \sqrt{2}$

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

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

Solution

$0$

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

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

Solution

343

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

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

Solution

$\frac{1}{w}$

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

In the following exercises, rationalize the denominator.

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

Solution

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

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

In the following exercises, solve.

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

Solution

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

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 hiker. Round your answer to the nearest tenth of a second.

Solution

6.1 seconds

rational exponents

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

Rational Exponents

Simplify Expressions with a1n

Solution

Solution

Solution

Solution

Solution

Solution

Simplify Expressions with amn

Solution

Solution

Solution

Solution

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

Solution

Solution

Solution

Solution

Solution

Solution

Key Concepts

Section Exercises

Practice Makes Perfect

Everyday Math

Writing Exercises

Chapter 9 Review Exercises

Practice Test

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

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