
# 9.8: Rational Exponents


Skills to Develop

By the end of this section, you will be able to:
• 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

Before you get started, take this readiness quiz.

1. Add: $$\frac{7}{15}+\frac{5}{12}$$.
If you missed this problem, review [link].
2. Simplify: $$(4x^{2}y^{5})^3$$.
If you missed this problem, review [link].
3. Simplify: $$5^{−3}$$.
If you missed this problem, review [link].

# 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 $$(a^m)^n=a^{m·n}$$ 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.

$\begin{array}{cc} {}&{(8^p)^3=8}\\ {\text{Multiply the exponents on the left.}}&{8^{3p}=8}\\ {\text{Write the exponent 1 on the right.}}&{8^{3p}=8^1}\\ {\text{The exponents must be equal.}}&{3p=1}\\ {\text{Solve for p.}}&{p=\frac{1}{3}}\\ \nonumber \end{array}$

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 $$a^{\frac{1}{n}}=\sqrt[n]{a}$$.

Definition: RATIONAL EXPONENT $$a^{\frac{1}{n}}$$

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

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 $$\PageIndex{1}$$

1. $$x^{\frac{1}{2}}$$
2. $$y^{\frac{1}{3}}$$
3. $$z^{\frac{1}{4}}$$.

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

 1. $$x^{\frac{1}{2}}$$ 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. $$\sqrt{x}$$ 2. $$y^{\frac{1}{3}}$$ The denominator of the exponent is 3, so the index is 3. $$\sqrt[3]{y}$$ 3. $$z^\frac{1}{4}}$$ The denominator of the exponent is 4, sothe index is 4. $$\sqrt[4]{z}$$

Example $$\PageIndex{2}$$

1. $$t^{\frac{1}{2}}$$
2. $$m^{\frac{1}{3}}$$
3. $$r^{\frac{1}{4}}$$.
1. $$\sqrt{t}$$
2. $$\sqrt[3]{m}$$
3. $$\sqrt[4]{r}$$

Example $$\PageIndex{3}$$

1. $$b^{\frac{1}{2}}$$
2. $$z^{\frac{1}{3}}$$
3. $$p^{\frac{1}{4}}$$.
1. $$\sqrt{b}$$
2. $$\sqrt[3]{z}$$
3. $$\sqrt[4]{p}$$

Example $$\PageIndex{4}$$

Write with a rational exponent:

1. $$\sqrt{x}$$
2. $$\sqrt[3]{y}$$
3. $$\sqrt[4]{z}$$.

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

 1. $$\sqrt{x}$$ No index is shown, so it is 2. The denominator of the exponent will be 2. $$x^{\frac{1}{2}}$$ 2. $$\sqrt[3]{y}$$ The index is 3, so the denominator of the exponent is 3. $$y^{\frac{1}{3}}$$ 3. $$\sqrt[4]{z}$$ The index is 4, so the denominator of the exponent is 4. $$z^{\frac{1}{4}}$$

Example $$\PageIndex{5}$$

Write with a rational exponent:

1. $$\sqrt{s}$$
2. $$\sqrt[3]{x}$$
3. $$\sqrt[4]{b}$$.
1. $$s^{\frac{1}{2}}$$
2. $$x^{\frac{1}{3}}$$
3. $$b^{\frac{1}{4}}\ Example \(\PageIndex{6}$$

Write with a rational exponent:

1. $$\sqrt{v}$$
2. $$\sqrt[3]{p}$$
3. $$\sqrt[4]{p}$$.
1. $$v^{\frac{1}{2}}$$
2. $$p^{\frac{1}{3}}$$
3. $$p^{\frac{1}{4}}$$

Example $$\PageIndex{7}$$

Write with a rational exponent:

1. $$\sqrt{5y}$$
2. $$\sqrt[3]{4x}$$
3. $$3\sqrt[4]{5z}$$.
 1. $$\sqrt{5y}$$ No index is shown, so it is 2. The denominator of the exponent will be 2. $$(5y)^{\frac{1}{2}}$$ 2. $$\sqrt[3]{4x}$$ The index is 3, so the denominator of the exponent is 3. $$(4x)^{\frac{1}{3}}$$ 3. $$3\sqrt[4]{5z}$$ The index is 4, so the denominator of the exponent is 4. $$3(5z)^{\frac{1}{4}}$$

Example $$\PageIndex{8}$$

Write with a rational exponent:

1. $$\sqrt{10m}$$
2. $$\sqrt[5]{3n}$$
3. $$3\sqrt[4]{6y}$$.
1. $$(10^m)^{\frac{1}{2}}$$
2. $$(3n)^{\frac{1}{5}}$$
3. $$(486y)^{\frac{1}{4}}$$

Example $$\PageIndex{9}$$

Write with a rational exponent:

1. $$\sqrt[7]{3k}$$
2. $$\sqrt[4]{5j}$$
3. $$\sqrt[3]{82a}$$.
1. $$(3k)^{\frac{1}{7}}$$
2. $$(5j)^{\frac{1}{4}}$$
3. $$(1024a)^{\frac{1}{3}}$$

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

Example $$\PageIndex{10}$$

Simplify:

1. $$25^{\frac{1}{2}}$$
2. $$64^{\frac{1}{3}}$$
3. $$256^{\frac{1}{4}}$$.
 1. $$25^{\frac{1}{2}}$$ Rewrite as a square root. $$\sqrt{25}$$ Simplify. 5 2. $$64^{\frac{1}{3}}$$ Rewrite as a cube root. $$\sqrt[3]{64}$$ Recognize 64 is a perfect cube. $$\sqrt[3]{4^3}$$ Simplify. 4 3. $$256^{\frac{1}{4}}$$ Rewrite as a fourth root. $$\sqrt[4]{256}$$ Recognize 256 is a perfect fourth power. $$\sqrt[4]{4^4}$$ Simplify. 4

Example $$\PageIndex{11}$$

Simplify:

1. $$36^{\frac{1}{2}}$$
2. $$8^{\frac{1}{3}}$$
3. $$16^{\frac{1}{4}}$$.
1. 6
2. 2
3. 2

Example $$\PageIndex{12}$$

Simplify:

1. $$100^{\frac{1}{2}}$$
2. $$27^{\frac{1}{3}}$$
3. $$81^{\frac{1}{4}}$$.
1. 10
2. 3
3. 3

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 $$\PageIndex{13}$$

Simplify:

1. $$(−64)^{\frac{1}{3}}$$
2. $$−64^{\frac{1}{3}}$$
3. $$(64)^{−\frac{1}{3}}$$.
 1. $$(−64)^{\frac{1}{3}}$$ Rewrite as a cube root. $$\sqrt[3]{−64}$$ Rewrite−64 as a perfect cube. $$\sqrt[3]{(−4)^3}$$ Simplify. −4 2. $$−64^{\frac{1}{3}}$$ 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 3. $$(64)^{−\frac{1}{3}}$$ Rewrite as a fraction with a positive exponent, using the property, $$a^{−n}=\frac{1}{a^n}$$. Write as a cube root. $$\frac{1}{\sqrt[3]{64}}$$ Rewrite 64 as $$4^3$$. $$\frac{1}{\sqrt[3]{4^3}}$$ Simplify. $$\frac{1}{4}$$

Example $$\PageIndex{14}$$

Simplify:

1. $$(−125)^{\frac{1}{3}}$$
2. $$−125^{\frac{1}{3}}$$
3. $$(125)^{−\frac{1}{3}}$$.
1. −5
2. −5
3. $$\frac{1}{5}$$

Example $$\PageIndex{15}$$

Simplify:

1. $$(−32)^{\frac{1}{5}}$$
2. $$−32^{\frac{1}{5}}$$
3. $$(32)^{−\frac{1}{5}}$$.
1. −2
2. −2
3. $$\frac{1}{2}$$

Example $$\PageIndex{16}$$

Simplify:

1. $$(−16)^{\frac{1}{4}}$$
2. $$−16^{\frac{1}{4}}$$
3. $$(16)^{−\frac{1}{4}}$$.
 1. $$(−16)^{\frac{1}{4}}$$ Rewrite as a fourth root. $$\sqrt[4]{−16}$$ There is no real number whose fourth power is −16. 2. $$−16^{\frac{1}{4}}$$ The exponent applies only to the 16. $$−(16^{\frac{1}{4}})$$ Rewrite as a fourth root. $$−\sqrt[4]{16}$$ Rewrite 16 as $$2^4$$ $$−\sqrt[4]{2^4}$$ Simplify. −2 3. $$(16)^{−\frac{1}{4}}$$ Rewrite as a fraction with a positive exponent, 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. $$\frac{1}{2}$$

Example $$\PageIndex{17}$$

Simplify:

1. $$(−64)^{\frac{1}{2}}$$
2. $$−64^{\frac{1}{2}}$$
3. $$(64)^{−\frac{1}{2}}$$.
1. −8
2. −8
3. $$\frac{1}{8}$$

Example $$\PageIndex{18}$$

Simplify:

1. $$(−256)^{\frac{1}{4}}$$
2. $$−256^{\frac{1}{4}}$$
3. $$(256)^{−\frac{1}{4}}$$.
1. −4
2. −4
3. $$\frac{1}{4}$$

# 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.

$\begin{array}{ll} {}&{(a^{\frac{1}{n}})^m}\\ {\text{Multiply the exponents.}}&{a^{\frac{1}{n}·m}}\\ {\text{Simplify.}}&{a^{\frac{m}{n}}}\\ {\text{So} a^{\frac{m}{n}}=(\sqrt[n]{a})^m \text{also.}}&{}\\ \nonumber \end{array}$

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

$\begin{array}{ll} {}&{(a^m)^{\frac{1}{n}}}\\ {\text{Multiply the exponents.}}&{a^{m·\frac{1}{n}}}\\ {\text{Simplify.}}&{a^{\frac{m}{n}}}\\ {\text{So} a^{\frac{m}{n}}=\sqrt[n]{a^m} \text{also.}}&{}\\ \nonumber \end{array}$

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.

Definition: RATIONAL EXPONENT $$a^{\frac{m}{n}}$$

For any positive integers m and n,

$$a^{\frac{m}{n}}=(\sqrt[n]{a})^m$$

$$a^{\frac{m}{n}}=\sqrt[n]{a^m}$$

Example $$\PageIndex{19}$$

Write with a rational exponent:

1. $$\sqrt{y^3}$$
2. $$\sqrt[3]{x^2}$$
3. $$\sqrt[4]{z^3}$$

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

Example $$\PageIndex{20}$$

Write with a rational exponent:

1. $$\sqrt{x^5}$$
2. $$\sqrt[4]{z^3}$$
3. $$\sqrt[5]{y^2}$$.
1. $$x^{\frac{5}{2}}$$
2. $$z^{\frac{3}{4}}$$
3. $$y^{\frac{2}{5}}$$

Example $$\PageIndex{21}$$

Write with a rational exponent:

1. $$\sqrt[5]{a^2}$$
2. $$\sqrt[3]{b^7}$$
3. $$\sqrt[4]{m^5}$$.
1. $$a^{\frac{2}{5}}$$
2. $$b^{\frac{7}{3}}$$
3. $$m^{\frac{5}{4}}$$

Example $$\PageIndex{22}$$

Simplify:

1. $$9^{\frac{3}{2}}$$
2. $$125^{\frac{2}{3}}$$
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.

 1. $$9^{\frac{3}{2}}$$ 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 2. $$125^{\frac{2}{3}}$$ The power of the radical is the numerator of the exponent, 2. Since the denominator of the exponent is 3, this is a square root. $$(\sqrt[3]{125})^2$$ Simplify. $$5^2$$ 25 3. $$81^{\frac{3}{4}}$$ The power of the radical is the numerator of the exponent, 2. Since the denominator of the exponent is 3, this is a square root. $$(\sqrt[4]{81})^3$$ Simplify. $$3^3$$ 27

Example $$\PageIndex{23}$$

Simplify:

1. $$4^{\frac{3}{2}}$$
2. $$27^{\frac{2}{3}}$$
3. $$625^{\frac{3}{4}}$$.
1. 8
2. 9
3. 125

Example $$\PageIndex{24}$$

Simplify:

1. $$8^{\frac{5}{3}}$$
2. $$81^{\frac{3}{2}}$$
3. $$16^{\frac{3}{4}}$$.
1. 32
2. 729
3. 8

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

Example $$\PageIndex{25}$$

Simplify:

1. $$16^{−\frac{3}{2}}$$
2. $$32^{−\frac{2}{5}}$$
3. $$4^{−\frac{5}{2}}$$

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

 1. $$16^{−\frac{3}{2}}$$ 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. $$\frac{1}{4^3}$$ $$\frac{1}{64}$$ 2. $$32^{−\frac{2}{5}}$$ 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. $$\frac{1}{2^2}$$ $$\frac{1}{4}$$ 3. $$4^{−\frac{5}{2}}$$ 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 $$\PageIndex{26}$$

Simplify:

1. $$8^{−\frac{5}{3}}$$8
2. $$81^{−\frac{3}{2}}$$
3. $$16^{−\frac{3}{4}}$$.
1. $$\frac{1}{32}$$
2. $$\frac{1}{729}$$
3. $$\frac{1}{8}$$

Example $$\PageIndex{27}$$

Simplify:

1. $$4^{−\frac{3}{2}}$$
2. $$27^{−\frac{2}{3}}$$
3. $$625^{−\frac{3}{4}}$$.
1. $$\frac{1}{8}$$
2. $$\frac{1}{9}$$
3. $$\frac{1}{125}$$

Example $$\PageIndex{28}$$

Simplify:

1. $$−25^{\frac{3}{2}}$$
2. $$−25^{−\frac{3}{2}}$$
3. $$(−25)^{\frac{3}{2}}$$.
 1. $$−25^{\frac{3}{2}}$$ Rewrite in radical form. $$−(\sqrt{25})^3$$ Simplify the radical $$−5^3$$ Simplify. −125 2. $$−25^{−\frac{3}{2}}$$ 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. $$−\frac{1}{125}$$ 3. $$(−25)^{\frac{3}{2}}$$. Rewrite in radical form. $$(\sqrt{−25})^3$$ There is no real number whose square root is−25. Not a real number.

Example $$\PageIndex{29}$$

Simplify:

1. $$−16^{\frac{3}{2}}$$
2. $$−16^{−\frac{3}{2}}$$
3. $$(−16)^{−\frac{3}{2}}$$.
1. −64
2. $$−\frac{1}{64}$$
3. not a real number

Example $$\PageIndex{30}$$

Simplify:

1. $$−81^{\frac{3}{2}}$$
2. $$−81^{−\frac{3}{2}}$$
3. $$(−81)^{−\frac{3}{2}}$$.
1. −729
2. $$−\frac{1}{729}$$
3. 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.

SUMMARY OF EXPONENT PROPERTIES

If a,b are real numbers and m,n are rational numbers, then

$\begin{array}{ll} {\textbf{Product Property}}&{a^m·a^n=a^{m+n}}\\ {\textbf{Power Property}}&{(a^m)^n=a^{m·n}}\\ {\textbf{Product to a Power}}&{(ab)^m=a^{m}b^{m}}\\ {\textbf{Quotient Property}}&{\frac{a^m}{a^n}=a^{m−n} , a \ne 0, m>n}\\ {}&{\frac{a^m}{a^n}=\frac{1}{a^{n−m}}, a \ne 0, n>m}\\ {\textbf{Zero Exponent Definition}}&{a^0=1, a \ne 0}\\ {\textbf{Quotient to a Power Property}}&{(\frac{a}{b})^m=\frac{a^m}{b^m}, b \ne 0}\\ \nonumber \end{array}$

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

Example $$\PageIndex{31}$$

Simplify:

1. $$2^{\frac{1}{2}}·2^{\frac{5}{2}}$$
2. $$x^{\frac{2}{3}}·x^{\frac{4}{3}}$$
3. $$z^{\frac{3}{4}}·z^{\frac{5}{4}}$$.
 1. $$2^{\frac{1}{2}}·2^{\frac{5}{2}}$$ 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 2. $$x^{\frac{2}{3}}·x^{\frac{4}{3}}$$ 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. $$x^2$$ 3. $$z^{\frac{3}{4}}·z^{\frac{5}{4}}$$ 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. $$z^2$$

Example $$\PageIndex{32}$$

Simplify:

1. $$3^{\frac{2}{3}}·3^{\frac{4}{3}}$$
2. $$y^{\frac{1}{3}}·y^{\frac{8}{3}}$$
3. $$m^{\frac{1}{4}}·m^{\frac{3}{4}}$$.
1. 9
2. $$y^3$$
3. m

Example $$\PageIndex{33}$$

Simplify:

1. $$5^{\frac{3}{5}}·5^{\frac{7}{5}}$$
2. $$z^{\frac{1}{8}}·z^{\frac{7}{8}}$$
3. $$n^{\frac{2}{7}}·n^{\frac{5}{7}}$$.
1. 25
2. z
3. n

We will use the Power Property in the next example.

Example $$\PageIndex{34}$$

Simplify:

1. $$(x^4)^{\frac{1}{2}}$$
2. $$(y^6)^{\frac{1}{3}}$$
3. $$(z^9)^{\frac{2}{3}}$$.
 1. $$(x^4)^{\frac{1}{2}}$$ To raise a power to a power, we multiply the exponents. $$x^{4·\frac{1}{2}}$$ Simplify. $$x^2$$ 2. $$(y^6)^{\frac{1}{3}}$$ To raise a power to a power, we multiply the exponents. $$y^{6·\frac{1}{3}}$$ Simplify. $$y^2$$ 3. $$(z^9)^{\frac{2}{3}}$$ To raise a power to a power, we multiply the exponents. $$z^{9·\frac{2}{3}}$$ Simplify. $$z^6$$

Example $$\PageIndex{35}$$

Simplify:

1. $$(p^{10})^{\frac{1}{5}}$$
2. $$(q^8)^{\frac{3}{4}}$$
3. $$(x^6)^{\frac{4}{3}}$$
1. $$p^$$
2. $$q^6$$
3. $$x^8$$

Example $$\PageIndex{36}$$

Simplify:

1. $$(r^6)^{\frac{5}{3}}$$
2. $$(s^{12})^{\frac{3}{4}}$$
3. $$(m^9)^{\frac{2}{9}}$$
1. $$r^{10}$$
2. $$s^9$$
3. $$m^2$$

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

Example $$\PageIndex{37}$$

Simplify:

1. $$\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$$
2. $$\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$$
3. $$\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}$$.
 1. $$\frac{x^{\frac{4}{3}}}{x^{\frac{1}{3}}}$$ To divide with the same base, we subtract the exponents. $$x^{\frac{4}{3}−\frac{1}{3}}$$ Simplify. x 2. $$\frac{y^{\frac{3}{4}}}{y^{\frac{1}{4}}}$$ To divide with the same base, we subtract the exponents. $$y^{\frac{3}{4}−\frac{1}{4}}$$ Simplify. $$y^{\frac{1}{2}}$$ 3. $$\frac{z^{\frac{2}{3}}}{z^{\frac{5}{3}}}$$ To divide with the same base, we subtract the exponents. $$z^{\frac{2}{3}−\frac{5}{3}}$$ Rewrite without a negative exponent. $$\frac{1}{z}$$

Example $$\PageIndex{38}$$

Simplify:

1. $$\frac{u^{\frac{5}{4}}}{u^{\frac{1}{4}}}$$
2. $$\frac{v^{\frac{3}{5}}}{v^{\frac{2}{5}}}$$
3. $$\frac{x^{\frac{2}{3}}}{x^{\frac{5}{3}}}$$.
1. u
2. $$v^{\frac{1}{5}}$$
3. $$\frac{1}{x}$$

Example $$\PageIndex{39}$$

Simplify:

1. $$\frac{c^{\frac{12}{5}}}{c^{\frac{2}{5}}}$$
2. $$\frac{m^{\frac{5}{4}}}{m^{\frac{9}{4}}}$$
3. $$\frac{d^{\frac{1}{5}}}{d^{\frac{6}{5}}}$$.
1. $$c^2$$
2. $$\frac{1}{m}$$
3. $$\frac{1}{d}$$

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 $$\PageIndex{40}$$

Simplify:

1. $$(27u^{\frac{1}{2}})^{\frac{2}{3}}$$
2. $$(8v^{\frac{1}{4}})^{\frac{2}{3}}$$.
 1. $$(27u^{\frac{1}{2}})^{\frac{2}{3}}$$ 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. $$9u^{\frac{1}{3}}$$ 2. $$(8v^{\frac{1}{4}})^{\frac{2}{3}}$$. 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. $$4v^{\frac{1}{6}}$$

Example $$\PageIndex{41}$$

Simplify:

1. $$32x^{\frac{1}{3}})^{\frac{3}{5}}$$
2. $$(64y^{\frac{2}{3}})^{\frac{1}{3}}$$.
1. $$8x^{\frac{1}{5}}$$
2. $$4y^{\frac{2}{9}}$$

Example $$\PageIndex{42}$$

Simplify:

1. $$(16m^{\frac{1}{3}})^{\frac{3}{2}}$$
2. $$(81n^{\frac{2}{5}})^{\frac{3}{2}}$$.
1. $$64m^{\frac{1}{2}}$$
2. $$729n^{\frac{3}{5}}$$

Example $$\PageIndex{43}$$

Simplify:

1. $$(m^{3}n^{9})^{\frac{1}{3}}$$
2. $$(p^{4}q^{8})^{\frac{1}{4}}$$.
 1. $$(m^{3}n^{9})^{\frac{1}{3}}$$ 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. $$mn^3$$ 2. $$(p^{4}q^{8})^{\frac{1}{4}}$$ 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. $$pq^2$$

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

Exercise $$\PageIndex{44}$$

Simplify:

1. $$\frac{x^{\frac{3}{4}}·x^{−\frac{1}{4}}}{x^{−\frac{6}{4}}}$$
2. $$\frac{y^{\frac{4}{3}}·y}{y^{−\frac{2}{3}}}$$.
 1. $$\frac{x^{\frac{3}{4}}·x^{−\frac{1}{4}}}{x^{−\frac{6}{4}}}$$ 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. $$x^2$$ 2. $$\frac{y^{\frac{4}{3}}·y}{y^{−\frac{2}{3}}}$$ 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. $$y^3$$

Example $$\PageIndex{45}$$

Simplify:

1. $$\frac{m^{\frac{2}{3}}·m^{−\frac{1}{3}}}{m^{−\frac{5}{3}}}$$
2. $$\frac{n^{\frac{1}{6}}·n}{n^{−\frac{11}{6}}}$$.
1. $$m^2$$
2. $$n^3$$

Example $$\PageIndex{46}$$

Simplify:

1. $$\frac{u^{\frac{4}{5}}·u^{−\frac{2}{5}}}{u^{−\frac{13}{5}}}$$
2. $$\frac{v^{\frac{1}{2}}·v}{v^{−\frac{7}{2}}}$$.
1. $$u^3$$
2. $$v^5$$

# Key Concepts

• Summary of Exponent Properties
• If a,b are real numbers and m,n are rational numbers, then
• Product Property $$a^m·a^n=a^{m+n}$$
• Power Property $$(a^m)^n=a^{m·n}$$
• Product to a Power $$(ab)^m=a^{m}b^{m}$$
• Quotient Property:

$$\frac{a^m}{a^n}=a^{m−n} , a \ne 0, m>n$$

$$\frac{a^m}{a^n}=\frac{1}{a^{n−m}}, a \ne 0, n>m$$

• Zero Exponent Definition $$a^0=1, a \ne 0$$
• Quotient to a Power Property $$(\frac{a}{b})^m=\frac{a^m}{b^m}, b \ne 0$$

# Section Exercises

## Practice Makes Perfect

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

In the following exercises, write as a radical expression.

Example $$\PageIndex{47}$$

1. $$x^{\frac{1}{2}}$$
2. $$y^{\frac{1}{3}}$$
3. $$z^{\frac{1}{4}}$$

Example $$\PageIndex{48}$$

1. $$r^{\frac{1}{2}}$$
2. $$s^{\frac{1}{3}}$$
3. $$t^{\frac{1}{4}}$$
1. $$\sqrt{r}$$
2. $$\sqrt[3]{s}$$
3. $$\sqrt[4]{t}$$

Example $$\PageIndex{49}$$

1. $$u^{\frac{1}{5}}$$
2. $$v^{\frac{1}{9}}$$
3. $$w^{\frac{1}{20}}$$

Example $$\PageIndex{50}$$

1. $$g^{\frac{1}{7}}$$
2. $$h^{\frac{1}{5}}$$
3. $$j^{\frac{1}{25}}$$
1. $$\sqrt[7]{g}$$
2. $$\sqrt[5]{h}$$
3. $$\sqrt[25]{j}$$
In the following exercises, write with a rational exponent.

Example $$\PageIndex{51}$$

1. $$−\sqrt[7]{x}$$
2. $$\sqrt[9]{y}$$
3. $$\sqrt[5]{f}$$

Example $$\PageIndex{52}$$

1. $$\sqrt[8]{r}$$
2. $$\sqrt[10]{s}$$
3. $$\sqrt[4]{t}$$
1. $$r^{\frac{1}{8}}$$
2. $$s^{\frac{1}{10}}$$
3. $$t^{\frac{1}{4}}$$

Example $$\PageIndex{53}$$

1. $$\sqrt[3]{a}\ 2. \(\sqrt[12]{b}$$
3. $$\sqrt{c}$$

Example $$\PageIndex{54}$$

1. $$\sqrt[5]{u}$$
2. $$\sqrt{v}$$

1. $$u^{\frac{1}{5}}$$
2. $$v^{\frac{1}{2}}$$
3. $$w^{\frac{1}{16}}$$

Example $$\PageIndex{55}$$

1. $$\sqrt[3]{7c}$$
2. $$\sqrt[7]{12d}$$
3. $$3\sqrt[4]{5f}$$

Example $$\PageIndex{56}$$

1. $$\sqrt[4]{5x}$$
2. $$\sqrt[8]{9y}$$
3. $$7\sqrt[5]{3z}$$
1. $$(5x)^{\frac{1}{4}}$$
2. $$(9y)^{\frac{1}{8}}$$
3. $$7(3z)^{\frac{1}{5}}$$

Example $$\PageIndex{57}$$

1. $$\sqrt{21p}$$
2. $$\sqrt[4]{8q}$$
3. $$\sqrt[6]{436r}$$

Example $$\PageIndex{58}$$

1. $$\sqrt[3]{25a}$$
2. $$\sqrt{3b}$$

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

In the following exercises, simplify.

Example $$\PageIndex{59}$$

1. $$81^{\frac{1}{2}}$$
2. $$125^{\frac{1}{3}}$$
3. $$64^{\frac{1}{2}}$$

Example $$\PageIndex{60}$$

1. $$625^{\frac{1}{4}}$$
2. $$243^{\frac{1}{5}}$$
3. $$32^{\frac{1}{5}}$$
1. 5
2. 3
3. 2

Example $$\PageIndex{61}$$

1. $$16^{\frac{1}{4}}$$
2. $$16^{\frac{1}{2}}$$
3. $$3125^{\frac{1}{5}}$$

Example $$\PageIndex{62}$$

1. $$216^{\frac{1}{3}}$$
2. $$32^{\frac{1}{5}}$$
3. $$81^{\frac{1}{4}}$$
1. 6
2. 2
3. 3

Example $$\PageIndex{63}$$

1. $$(−216)^{\frac{1}{3}}$$
2. $$−216^{\frac{1}{3}}$$
3. $$(216)^{−\frac{1}{3}}$$

Example $$\PageIndex{64}$$

1. $$(−243)^{\frac{1}{5}}$$
2. $$−243^{\frac{1}{5}}$$
3. $$(243)^{−\frac{1}{5}}$$
1. −3
2. −3
3. $$\frac{1}{3}$$

Example $$\PageIndex{65}$$

1. $$(−1)^{\frac{1}{3}}$$
2. $$−1^{\frac{1}{3}}$$
3. $$(1)^{−\frac{1}{3}}$$

Example $$\PageIndex{66}$$

1. $$(−1000)^{\frac{1}{3}}$$
2. $$−1000^{\frac{1}{3}}$$
3. $$(1000)^{−\frac{1}{3}}$$
1. −10
2. −10
3. $$\frac{1}{10} Example \(\PageIndex{67}$$

1. $$(−81)^{\frac{1}{4}}$$
2. $$−81^{\frac{1}{4}}$$
3. $$(81)^{−\frac{1}{4}}$$

Example $$\PageIndex{68}$$

1. $$(−49)^{\frac{1}{2}}$$
2. $$−49^{\frac{1}{2}}$$
3. $$(49)^{−\frac{1}{2}}$$
1. not a real number
2. −7
3. $$\frac{1}{7}$$

Example $$\PageIndex{69}$$

1. $$(−36)^{\frac{1}{2}}$$
2. $$−36^{\frac{1}{2}}$$
3. $$(36)^{−\frac{1}{2}}$$

Example $$\PageIndex{70}$$

1. $$(−1)^{\frac{1}{4}}$$
2. $$(1)^{−\frac{1}{4}}$$
3. $$−1^{\frac{1}{4}}$$
1. not a real number
2. 1
3. −1

Example $$\PageIndex{71}$$

1. $$(−100)^{\frac{1}{2}}$$
2. $$−100^{\frac{1}{2}}$$
3. $$(100)^{−\frac{1}{2}}$$

Example $$\PageIndex{72}$$

1. $$(−32)^{\frac{1}{5}}$$
2. $$(243)^{−\frac{1}{5}}$$
3. $$−125^{\frac{1}{3}}$$
1. −2
2. $$\frac{1}{3}$$
3. −5

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

In the following exercises, write with a rational exponent.

Example $$\PageIndex{73}$$

1. $$\sqrt{m^5}$$
2. $$\sqrt[3]{n^2}$$
3. $$\sqrt[4]{p^3}$$

Example $$\PageIndex{74}$$

1. $$\sqrt[4]{r^7}$$
2. $$\sqrt[5]{s^3}$$
3. $$\sqrt[3]{t^7}$$
1. $$r^{\frac{7}{4}}$$
2. $$s^{\frac{3}{5}}$$
3. $$t^{\frac{7}{3}}$$

Exercise $$\PageIndex{75}$$

1. $$\sqrt[5]{u^2}$$
2. $$\sqrt[5]{v^8}$$
3. $$\sqrt[9]{w^4}$$

Example $$\PageIndex{76}$$

1. $$\sqrt[3]{a}$$
2. $$\sqrt[5]{b}$$
3. $$\sqrt[3]{c^5}$$
1. $$a^{\frac{1}{3}}$$
2. $$b^{\frac{1}{5}}$$
3. $$c^{\frac{5}{3}}$$
In the following exercises, simplify.

Example $$\PageIndex{77}$$

1. $$16^{\frac{3}{2}}$$
2. $$8^{\frac{2}{3}}$$
3. $$10,000^{\frac{3}{4}}$$

Example $$\PageIndex{78}$$

1. $$1000^{\frac{2}{3}}$$
2. $$25^{\frac{3}{2}}$$
3. $$32^{\frac{3}{5}}$$
1. 100
2. 125
3. 8

Example $$\PageIndex{79}$$

1. $$275^{\frac{3}{2}}$$
2. $$16^{\frac{5}{4}}$$
3. $$32^{\frac{2}{5}}$$

Example $$\PageIndex{80}$$

1. $$16^{\frac{3}{2}}$$
2. $$125^{\frac{5}{3}}$$
3. $$64^{\frac{4}{3}}$$
1. 64
2. 3125
3. 256

Example $$\PageIndex{81}$$

1. $$32^{\frac{2}{5}}$$
2. $$27^{−\frac{2}{3}}$$
3. $$25^{−\frac{3}{2}}$$

Example $$\PageIndex{82}$$

1. $$64^{\frac{5}{2}}$$
2. $$81^{−\frac{3}{2}}$$
3. $$27^{−\frac{4}{3}}$$
1. 32,768
2. $$\frac{1}{729}$$
3. $$\frac{1}{81}$$

Example $$\PageIndex{83}$$

1. $$25^{\frac{3}{2}}$$
2. $$9^{−\frac{3}{2}}$$
3. $$(−64)^{\frac{2}{3}}$$

Example $$\PageIndex{84}$$

1. $$100^{\frac{3}{2}}$$
2. $$49^{−\frac{5}{2}}$$
3. $$(−100)^{\frac{3}{2}}$$
1. 1000
2. $$\frac{1}{16,807}$$
3. not a real numbe

Example $$\PageIndex{85}$$

1. $$−9^{\frac{3}{2}}$$
2. $$−9^{−\frac{3}{2}}$$
3. $$(−9)^{\frac{3}{2}}$$

Example $$\PageIndex{86}$$

1. $$−64^{\frac{3}{2}}$$
2. $$−64^{−\frac{3}{2}}$$
3. $$(−64)^{\frac{3}{2}}$$
1. −512
2. $$−\frac{1}{512}$$
3. not a real number

Example $$\PageIndex{87}$$

1. $$−100^{\frac{3}{2}}$$
2. $$−100^{−\frac{3}{2}}$$
3. $$(−100)^{\frac{3}{2}}$$

Example $$\PageIndex{88}$$

1. $$−49^{\frac{3}{2}}$$
2. $$−49^{−\frac{3}{2}}$$
3. $$(−49)^{\frac{3}{2}}$$
1. −343
2. $$−\frac{1}{343}$$
3. not a real number

Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Example $$\PageIndex{89}$$

1. $$4^{\frac{5}{8}}·4^{\frac{11}{8}}$$
2. $$m^{\frac{7}{12}}·m^{\frac{17}{12}}$$
3. $$p^{\frac{3}{7}}·p^{\frac{18}{7}}$$

Example $$\PageIndex{90}$$

1. $$6^{\frac{5}{2}}·6^{\frac{12}{6}}$$
2. $$n^{\frac{2}{10}}·n^{\frac{8}{10}}$$
3. $$q^{\frac{2}{5}}·q^{\frac{13}{5}}$$
1. 216
2. n
3. $$q^3$$

Example $$\PageIndex{91}$$

1. $$5^{\frac{1}{2}}·5^{\frac{7}{2}}$$
2. $$c^{\frac{3}{4}}·c^{\frac{9}{4}}$$
3. $$d^{\frac{3}{5}}·d^{\frac{2}{5}}$$

Example $$\PageIndex{92}$$

1. $$10^{\frac{1}{3}}·10^{\frac{5}{3}}$$
2. $$x^{\frac{5}{6}}·x^{\frac{7}{6}}$$
3. $$y^{\frac{11}{8}}·y^{\frac{21}{8}}$$
1. 100
2. $$x^2$$
3. $$y^4$$

Example $$\PageIndex{93}$$

1. $$(m^6)^{\frac{5}{2}}$$
2. $$(n^9)^{\frac{4}{3}}$$
3. $$(p^{12})^{\frac{3}{4}}$$

Example $$\PageIndex{94}$$

1. $$(a^{12})^{\frac{1}{6}}$$
2. $$(b^{15})^{\frac{3}{5}}$$
3. $$(c^{11})^{\frac{1}{11}}$$
1. $$a^2$$
2. $$b^9$$
3. c

Example $$\PageIndex{95}$$

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

Example $$\PageIndex{96}$$

1. $$(h^6)^{\frac{4}{3}}$$
2. $$(k^{12})^{\frac{3}{4}}$$
3. $$(j^{10})^{\frac{7}{5}}$$
1. $$h^8$$
2. $$k^9$$
3. $$j^{14}$$

Example $$\PageIndex{97}$$

1. $$\frac{x^{\frac{7}{2}}}{x^{\frac{5}{2}}}$$
2. $$\frac{y^{\frac{5}{2}}}{y^{\frac{1}{2}}}$$
3. $$\frac{r^{\frac{4}{5}}}{r^{\frac{9}{5}}}$$

Example $$\PageIndex{98}$$

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

Example $$\PageIndex{99}$$

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

Example $$\PageIndex{100}$$

1. $$\frac{u^{\frac{13}{9}}}{u^{\frac{4}{9}}}$$
2. $$\frac{r^{\frac{15}{7}}}{r^{\frac{8}{7}}}$$
3. $$\frac{n^{\frac{3}{5}}}{n^{\frac{8}{5}}}$$
1. u
2. r
3. $$\frac{1}{n}$$

Example $$\PageIndex{101}$$

1. $$(9p^{\frac{2}{3}})^{\frac{5}{2}}$$
2. $$(27q^{\frac{3}{2}})^{\frac{4}{3}}$$​​​​​

Example $$\PageIndex{102}$$

1. $$(81r^{\frac{4}{5}})^{\frac{1}{4}}$$
2. $$(64s^{\frac{3}{7}})^{\frac{1}{6}}$$
1. $$3r^{\frac{1}{5}}$$
2. $$2s^{\frac{1}{14}}$$

Example $$\PageIndex{103}$$

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

Example $$\PageIndex{104}$$

1. $$(27m^{\frac{3}{4}})^{\frac{2}{3}}$$
2. $$(625n^{\frac{8}{3}})^{\frac{3}{4}}$$
1. $$9m^{\frac{1}{2}}$$
2. $$125n^2$$

Example $$\PageIndex{105}$$

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

Example $$\PageIndex{106}$$

1. $$(r^{8}s^{4})^{\frac{1}{4}}$$
2. $$(u^{15}v^{20})^{\frac{1}{5}}$$
1. $$r^{2}s$$
2. $$u^{3}v^{4}$$

Example $$\PageIndex{107}$$

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

Example $$\PageIndex{108}$$

1. $$(r^{16}s^{10})^{\frac{1}{2}}$$
2. $$(u^{10}v^5)^{\frac{4}{5}}$$
1. $$r^{8}s^{5}$$
2. $$u^{8}v^{4}$$

Example $$\PageIndex{109}$$

1. $$\frac{r^{\frac{5}{2}}·r^{−\frac{1}{2}}}{r^{−\frac{3}{2}}}$$
2. $$\frac{s^{\frac{1}{5}}·s}{s^{−\frac{9}{5}}}$$

Example $$\PageIndex{110}$$

1. $$\frac{a^{\frac{3}{4}}·a^{−\frac{1}{4}}}{a^{−\frac{10}{4}}}$$
2. $$\frac{b^{\frac{2}{3}}·b}{b^{−\frac{7}{3}}}$$
1. $$a^3$$
2. $$b^4$$

Example $$\PageIndex{111}$$

1. $$\frac{c^{\frac{5}{3}}·c^{−\frac{1}{3}}}{c^{−\frac{2}{3}}}$$
2. $$\frac{d^{\frac{3}{5}}·d}{d^{−\frac{2}{5}}}$$

Example $$\PageIndex{112}$$

1. $$\frac{m^{\frac{7}{4}}·m^{−\frac{5}{4}}}{m^{−\frac{2}{4}}}$$
2. $$\frac{n^{\frac{3}{7}}·n}{n^{−\frac{4}{7}}}$$
1. m
2. $$n^2$$

Example $$\PageIndex{113}$$

$$4^{\frac{5}{2}}·4^{\frac{1}{2}}$$

Example $$\PageIndex{114}$$

$$n^{\frac{2}{6}}·n^{\frac{4}{6}}$$

n

Example $$\PageIndex{115}$$

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

Example $$\PageIndex{116}$$

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

$$b^6$$

Example $$\PageIndex{117}$$

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

Example $$\PageIndex{118}$$

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

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

Example $$\PageIndex{119}$$

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

Example $$\PageIndex{120}$$

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

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

Example $$\PageIndex{121}$$

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

Example $$\PageIndex{122}$$

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

$$u^{2}v^{3}$$​​​​​​

## Everyday Math

Example $$\PageIndex{123}$$

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.

Example $$\PageIndex{124}$$

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

Example $$\PageIndex{125}$$

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.

Example $$\PageIndex{126}$$

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

Example $$\PageIndex{127}$$

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

Example $$\PageIndex{128}$$

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

# Chapter 9 Review Exercises

## Simplify and Use Square Roots

Simplify Expressions with Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{1}$$

$$\sqrt{64}$$

Exercise $$\PageIndex{2}$$

$$\sqrt{144}$$

12

Exercise $$\PageIndex{3}$$

$$−\sqrt{25}$$

Exercise $$\PageIndex{4}$$

$$−\sqrt{81}$$

−9

Exercise $$\PageIndex{5}$$

$$\sqrt{−9}$$

Exercise $$\PageIndex{6}$$

$$\sqrt{−36}$$

not a real number

Exercise $$\PageIndex{7}$$

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

Exercise $$\PageIndex{8}$$

$$\sqrt{64+225}$$

17

Estimate Square Roots

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

Exercise $$\PageIndex{9}$$

$$\sqrt{28}$$

Exercise $$\PageIndex{10}$$

$$\sqrt{155}$$

$$12<\sqrt{155}<13$$

Approximate Square Roots

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

Exercise $$\PageIndex{11}$$

$$\sqrt{15}$$

Example $$\PageIndex{12}$$

$$\sqrt{57}$$

7.55

Simplify Variable Expressions with Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{13}$$

$$\sqrt{q^2}$$

Exercise $$\PageIndex{14}$$

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

8b

Exercis\e $$\PageIndex{15}$$

$$−\sqrt{121a^2}$$

Exercise $$\PageIndex{16}$$

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

15mn

Exercise $$\PageIndex{17}$$

$$−\sqrt{100q^2}$$

Exercise $$\PageIndex{18}$$

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

7y

Exercise $$\PageIndex{19}$$

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

Exercise $$\PageIndex{20}$$

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

11cd

## Simplify Square Roots

Use the Product Property to Simplify Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{21}$$

$$\sqrt{300}$$

Exercise $$\PageIndex{22}$$

$$\sqrt{98}$$

$$7\sqrt{2}$$

Exercise $$\PageIndex{23}$$

$$\sqrt{x^{13}}$$

Exercise $$\PageIndex{24}$$

$$\sqrt{y^{19}}$$

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

Exercise $$\PageIndex{25}$$

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

Exercise $$\PageIndex{26}$$

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

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

Exercise $$\PageIndex{27}$$

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

Exercise $$\PageIndex{28}$$

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

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

Exercise $$\PageIndex{29}$$

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

Exercise $$\PageIndex{30}$$

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

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

Exercise $$\PageIndex{31}$$

$$\frac{10−\sqrt{50}}{5}$$

Exercise $$\PageIndex{32}$$

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

$$1+\sqrt{2}$$

Use the Quotient Property to Simplify Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{33}$$

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

Exercise $$\PageIndex{34}$$

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

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

Exercise $$\PageIndex{35}$$

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

Exercise $$\PageIndex{36}$$

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

$$y^2$$

Exercise $$\PageIndex{37}$$

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

Exercise $$\PageIndex{38}$$

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

$$6q^2$$

Exercise $$\PageIndex{39}$$

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

Exercise $$\PageIndex{40}$$

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

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

Exercise $$\PageIndex{41}$$

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

Exercise $$\PageIndex{42}$$

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

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

Exercise $$\PageIndex{43}$$

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

Exercise $$\PageIndex{44}$$

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

$$\frac{2rs^3\sqrt{r}}{5}$$

## Add and Subtract Square Roots

Add and Subtract Like Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{45}$$

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

Exercise $$\PageIndex{46}$$

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

$$12\sqrt{5}$$

Exercise $$\PageIndex{47}$$

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

Exercise $$\PageIndex{48}$$

$$6\sqrt{m}−2\sqrt{m}$$

$$4\sqrt{m}$$

Exercise $$\PageIndex{49}$$

$$−3\sqrt{7}+2\sqrt{7}−\sqrt{7}$$

Exercise $$\PageIndex{50}$$

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

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

Exercise $$\PageIndex{51}$$

$$3\sqrt{5xy}−\sqrt{5xy}+3\sqrt{5xy}$$

Exercise $$\PageIndex{52}$$

$$2\sqrt{3rs}+\sqrt{3rs}−5\sqrt{rs}$$

$$3\sqrt{3rs}−5\sqrt{rs}$$

Add and Subtract Square Roots that Need Simplification

In the following exercises, simplify.

Exercise $$\PageIndex{53}$$

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

Exercise $$\PageIndex{54}$$

$$\sqrt{8}+\sqrt{32}$$

$$5\sqrt{2}$$

Exercise $$\PageIndex{55}$$

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

Exercise $$\PageIndex{56}$$

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

$$9\sqrt{3}$$

Exercise $$\PageIndex{57}$$

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

Exercise $$\PageIndex{58}$$

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

0

Exercise $$\PageIndex{59}$$

$$\sqrt{50y^5}−\sqrt{72y^5}$$

Exercise $$\PageIndex{60}$$

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

Multiply Square Roots

Multiply Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{61}$$

$$\sqrt{2}·\sqrt{20}$$

Exercise $$\PageIndex{62}$$

$$2\sqrt{2}·6\sqrt{14}$$

$$24\sqrt{7}$$

Exercise $$\PageIndex{63}$$

$$\sqrt{2m^2}·\sqrt{20m^4}$$

Exercise $$\PageIndex{64}$$

$$(\sqrt{62y})(\sqrt{350y^3})$$

$$180y^2$$

Exercise $$\PageIndex{65}$$

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

Exercise $$\PageIndex{66}$$

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

8

Exercise $$\PageIndex{67}$$

$$(−\sqrt{10})^2$$

Exercise $$\PageIndex{68}$$

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

50

Exercise $$\PageIndex{69}$$

$$(−3\sqrt{3})(5\sqrt{18})$$

​​​​​​​Use Polynomial Multiplication to Multiply Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{70}$$

$$10(2−\sqrt{7})$$

$$20−10\sqrt{7}$$

Exercise $$\PageIndex{71}$$

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

Exercise $$\PageIndex{72}$$

$$(5+\sqrt{2})(3−\sqrt{2})$$

$$13−2\sqrt{2}$$

Exercise $$\PageIndex{73}$$

$$(5−3\sqrt{7})(1−2\sqrt{7})$$

Exercise $$\PageIndex{74}$$

$$(1−3\sqrt{x})(5+2\sqrt{x})$$

$$5−13\sqrt{x}−6x$$

Exercise $$\PageIndex{75}$$

$$(3+4\sqrt{y})(10−\sqrt{y})$$

Exercise $$\PageIndex{76}$$

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

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

Exercise $$\PageIndex{77}$$

$$(2−6\sqrt{5})^2$$

Exercise $$\PageIndex{78}$$

$$(3+2\sqrt{7})(3−2\sqrt{7})$$

−19

Exercise $$\PageIndex{79}$$

$$(6−\sqrt{11})(6+\sqrt{11})$$

## Divide Square Roots

Divide Square Roots

In the following exercises, simplify.

Exercise $$\PageIndex{80}$$

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

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

Exercise $$\PageIndex{81}$$

$$\frac{2−\sqrt{12}}{6}$$

Exercise $$\PageIndex{82}$$

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

$$\frac{4}{3}$$

Exercise $$\PageIndex{83}$$

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

Exercise $$\PageIndex{84}$$

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

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

Exercise $$\PageIndex{85}$$

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

​​​​​​​Rationalize a One Term Denominator

In the following exercises, rationalize the denominator.

Exercise $$\PageIndex{86}$$

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

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

Exercise $$\PageIndex{87}$$

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

Exercise $$\PageIndex{88}$$

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

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

Exercise $$\PageIndex{89}$$

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

Exercise $$\PageIndex{90}$$

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

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

Exercise $$\PageIndex{91}$$

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

​​​​​​​Rationalize a Two Term Denominator

In the following exercises, rationalize the denominator.

Exercise $$\PageIndex{92}$$

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

$$\frac{16−12\sqrt{3}}{−11}$$

Exercise $$\PageIndex{93}$$

$$\frac{5}{2−\sqrt{10}}$$

Exercise $$\PageIndex{94}$$

$$\frac{4}{2−\sqrt{5}}$$

$$−8−4\sqrt{5}$$

Exercise $$\PageIndex{95}$$

$$\frac{5}{4−\sqrt{8}}$$

Exercise $$\PageIndex{96}$$

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

$$\frac{\sqrt{2p}−\sqrt{6}}{p−3}$$

Exercise $$\PageIndex{97}$$

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

## Solve Equations with Square Roots

In the following exercises, solve the equation.

Exercise $$\PageIndex{98}$$

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

5

Exercise $$\PageIndex{99}$$

$$\sqrt{4u−2}−4=0$$

Exercise $$\PageIndex{100}$$

$$\sqrt{6m+4}−5=0$$

$$\frac{7}{2}$$

Exercise $$\PageIndex{101}$$

$$\sqrt{2u−3}+2=0$$

Exercise $$\PageIndex{102}$$

$$\sqrt{u−4}+4=u$$

no solution

Exercise $$\PageIndex{103}$$

$$\sqrt{v−9}+9=0$$​​​​​​​

Exercise $$\PageIndex{104}$$

$$\sqrt{r−4}−r=−10$$

13

Exercise $$\PageIndex{105}$$

$$\sqrt{s−9}−s=−9$$

Exercise $$\PageIndex{106}$$

$$2\sqrt{2x−7}−4=8$$

$$\frac{43}{2}$$

Exercise $$\PageIndex{107}$$

$$\sqrt{2−x}=\sqrt{2x−7}$$

Exercise $$\PageIndex{108}$$

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

0

Exercise $$\PageIndex{109}$$

$$\sqrt{r}+3=\sqrt{r+4}$$​​​​​​​

Exercise $$\PageIndex{110}$$

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

$$\frac{11}{6}$$

Exercise $$\PageIndex{111}$$

$$\sqrt{n+11}−1=\sqrt{n+4}$$

Exercise $$\PageIndex{112}$$

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

11

​​​​​​​Use Square Roots in Applications

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

Exercise $$\PageIndex{113}$$

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.​​​​​​​

Exercise $$\PageIndex{114}$$

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

Exercise $$\PageIndex{115}$$

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 $$s=\sqrt{24d}$$ to find the speed of the car before the brakes were applied.

## Higher Roots

Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise $$\PageIndex{116}$$

1. $$\sqrt[6]{64}$$
2. $$\sqrt[3]{64}$$
1. 2
2. 4

Exercise $$\PageIndex{117}$$

1. $$\sqrt[3]{−27}$$
2. $$\sqrt[4]{−64}$$

Exercise $$\PageIndex{118}$$

1. $$\sqrt[9]{d^9}$$
2. $$\sqrt[8]{v^8}$$
1. d
2. |v|

Exercise $$\PageIndex{119}$$

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

Exercise $$\PageIndex{120}$$

1. $$\sqrt[4]{16x^8}$$
2. $$\sqrt[6]{64y^{12}}$$
1. $$2x^2$$
2. $$2y^2$$

Exercise $$\PageIndex{121}$$

1. $$\sqrt[7]{128r^{14}}$$
2. $$\sqrt[4]{81s^{24}}$$

Use the Product Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise $$\PageIndex{122}$$

1. $$\sqrt[9]{d^9}$$

1. d

​​​​​​​​​​​​​​

Exercise $$\PageIndex{123}$$

1. $$\sqrt[3]{54}$$
2. $$\sqrt[4]{128}$$

Exercise $$\PageIndex{124}$$

1. $$\sqrt[5]{64c^8}$$
2. $$\sqrt[4]{48d^7}$$
1. $$2c\sqrt[5]{2c^3}$$
2. $$2d\sqrt[4]{3d^3}$$

Exercise $$\PageIndex{125}$$

1. $$\sqrt[3]{343q^7}$$
2. $$\sqrt[6]{192r^9}$$

Exercise $$\PageIndex{126}$$

1. $$\sqrt[3]{−500}$$
2. $$\sqrt[4]{−16}$$
1. $$−5\sqrt[3]{4}$$
2. not a real number
​​​​​​​Use the Quotient Property to Simplify Expressions with Higher Roots

In the following exercises, simplify.

Exercise $$\PageIndex{127}$$

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

Exercise $$\PageIndex{128}$$

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

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

Exercise $$\PageIndex{129}$$

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

Exercise $$\PageIndex{130}$$

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

$$3z^2$$

Exercise $$\PageIndex{131}$$

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

In the following exercises, simplify.

Exercise $$\PageIndex{132}$$

$$4\sqrt[5]{20}−2\sqrt[5]{20}$$

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

Exercise $$\PageIndex{133}$$

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

Exercise $$\PageIndex{134}$$

$$\sqrt[4]{1250}−\sqrt[4]{162}$$

$$2\sqrt[4]{2}$$

Exercise $$\PageIndex{135}$$

$$\sqrt[3]{640c^5}−\sqrt[3]{−80c^3}$$

Exercise $$\PageIndex{136}$$

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

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

## Rational Exponents

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

In the following exercises, write as a radical expression.

Exercise $$\PageIndex{137}$$

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

Exercise $$\PageIndex{138}$$

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

​​​​​​​In the following exercises, write with a rational exponent.

Exercise $$\PageIndex{139}$$

$$\sqrt[5]{u}$$

Exercise $$\PageIndex{140}$$

$$\sqrt[6]{v}$$

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

Exercise $$\PageIndex{141}$$

$$\sqrt[3]{9m}$$

Exercise $$\PageIndex{142}$$

$$\sqrt[6]{10z}$$

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

​​​​​​​In the following exercises, simplify.

Exercise $$\PageIndex{143}$$

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

Exercise $$\PageIndex{144}$$

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

2

Exercise $$\PageIndex{145}$$

$$(−125)^{\frac{1}{3}}$$

Exercise $$\PageIndex{146}$$

$$(125)^{−\frac{1}{3}}$$

$$\frac{1}{5}$$

Exercise $$\PageIndex{147}$$

$$(−9)^{\frac{1}{2}}$$

Exercise $$\PageIndex{148}$$

$$(36)^{−\frac{1}{2}}$$

$$\frac{1}{6}$$

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

In the following exercises, write with a rational exponent.

Exercise $$\PageIndex{149}$$

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

Exercise $$\PageIndex{150}$$

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

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

In the following exercises, simplify.

Exercise $$\PageIndex{151}$$

$$27^{−\frac{2}{3}}$$

Exercise $$\PageIndex{152}$$

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

32,768

Exercise $$\PageIndex{153}$$

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

Exercise $$\PageIndex{154}$$

$$81^{−\frac{5}{2}}$$

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

​​​​​​​Use the Laws of Exponents to Simplify Expressions with Rational Exponents

In the following exercises, simplify.

Exercise $$\PageIndex{155}$$

$$3^{\frac{4}{5}}·3^{\frac{6}{5}}$$

Exercise $$\PageIndex{156}$$

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

$$x^8$$

Exercise $$\PageIndex{157}$$

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

Exercise $$\PageIndex{158}$$

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

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

Exercise $$\PageIndex{159}$$

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

Exercise $$\PageIndex{160}$$

$$\frac{z^{\frac{2}{3}}·z^{−\frac{1}{3}}}{z^{−\frac{5}{3}}}$$

$$z^2$$​​​​​​​

# Practice Test

In the following exercises, simplify.

Exercise $$\PageIndex{161}$$

$$\sqrt{81+144}$$

Exercise $$\PageIndex{162}$$

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

$$13m^{2}|n|$$

Exercise $$\PageIndex{163}$$

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

Exercise $$\PageIndex{164}$$

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

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

Exercise $$\PageIndex{165}$$

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

Exercise $$\PageIndex{166}$$

$$(3\sqrt{6y})(\sqrt{250y^3})$$

$$180y^2\sqrt{3}$$

Exercise $$\PageIndex{167}$$

$$(2−5\sqrt{x})(3+\sqrt{x})$$

Exercise $$\PageIndex{168}$$

$$(1−2\sqrt{q})^2$$

$$1−4\sqrt{q}+4q$$

Exercise $$\PageIndex{169}$$

1. $$\sqrt{a^{12}}$$
2. $$\sqrt[3]{b^{21}}$$

Exercise $$\PageIndex{170}$$

1. $$\sqrt[4]{81x^{12}}$$
2. $$\sqrt[6]{64y^{18}}$$
1. $$3x^3$$
2. $$2y^3$$

Exercise $$\PageIndex{171}$$

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

Exercise $$\PageIndex{172}$$

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

$$y\sqrt{2}$$

Exercise $$\PageIndex{173}$$

$$\frac{\sqrt{256x^7}}{\sqrt{54x^2}}$$

Exercise $$\PageIndex{174}$$

$$\sqrt[4]{512}−2\sqrt[4]{32}$$

0

Exercise $$\PageIndex{175}$$

1. $$256^{\frac{1}{4}}$$
2. $$243^{\frac{1}{5}}$$

Exercise $$\PageIndex{176}$$

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

343

Exercise $$\PageIndex{177}$$

$$25^{−\frac{5}{2}}$$

Exercise $$\PageIndex{178}$$

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

$$\frac{1}{w}$$

Exercise $$\PageIndex{179}$$

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

​​​​​​​In the following exercises, rationalize the denominator.

Exercise $$\PageIndex{180}$$

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

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

Exercise $$\PageIndex{181}$$

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

In the following exercises, solve.

Exercise $$\PageIndex{182}$$

$$3\sqrt{2x−3}−20=7$$

42​​​​​​​

Exercise $$\PageIndex{183}$$

$$\sqrt{3u−2}=\sqrt{5u+1}$$

​​​​​​​In the following exercise, solve.

Exercise $$\PageIndex{184}$$

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.

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