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# 6.5: Fractional Exponents

Difficulty Level: Basic Created by: CK-12
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Practice Fractional Exponents
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If an exponent usually tells you the number of times to multiply the base by itself, what does it mean if the exponent is a fraction? How can you think about and calculate 432\begin{align*}4^{\frac{3}{2}}\end{align*}?

### Guidance

A fraction exponent is related to a root. Raising a number to the power of 12\begin{align*}\frac{1}{2}\end{align*} is the same as taking the square root of the number. If you have amn\begin{align*}a^{\frac{m}{n}}\end{align*}, you can think about this expression in multiple ways:

amn=(am)1noramn=(a1n)m\begin{align*}a^{\frac{m}{n}}=(a^m)^{{\color{red}\frac{1}{n}}} \quad \text{or} \quad a^{\frac{m}{n}}=\left(a^{{\color{red}\frac{1}{n}}}\right)^m \end{align*}

amn=amnoramn=(an)m\begin{align*} a^{\frac{m}{n}}=\sqrt[{\color{red}n}]{a^{\color{red}m}} \quad \text{or} \quad a^{\frac{m}{n}}=(\sqrt[{\color{red}n}]{a})^{\color{red}m}\end{align*}

All of these ideas can be summarized as the following rule for fractional exponents:

amn=amn=(an)m m,nεN\begin{align*}\boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}=(\sqrt[n]{a})^m \ m, n \varepsilon N}\end{align*}

#### Example A

Simplify the following:

(125)23\begin{align*}(125)^{-\frac{2}{3}}\end{align*}

Solution:

(125)231125231(1253)2Apply the law of exponents for negative exponents am=1am.Apply the law of exponents for rational exponents amn=amn=(an)m m,nεN.\begin{align*}& (125)^{-\frac{2}{3}} && \text{Apply the law of exponents for negative exponents} \ \boxed{a^{-m}=\frac{1}{a^m}}.\\ & \frac{1}{125^{{\color{red}\frac{2}{3}}}} && \text{Apply the law of exponents for rational exponents} \ \boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left( \sqrt[n]{a}\right)^{m} \ m,n \varepsilon N.}\\ & \frac{1}{\left(\sqrt[{\color{red}3}]{125}\right)^{\color{red}2}}\end{align*}

The cube root of 125 is ‘5’.

152\begin{align*}\frac{1}{{\color{red}5}^2}\end{align*}

Evaluate the denominator.

125(125)23=125am=1am\begin{align*}& {\color{red}\frac{1}{25}}\\ & \boxed{(125)^{-\frac{2}{3}}=\frac{1}{25}}\\ & \boxed{a^{-m}=\frac{1}{a^m}}\end{align*}

#### Example B

Simplify the following:

(2a2b4)32\begin{align*}(2a^2b^4)^{\frac{3}{2}}\end{align*}

Solution:

(2a2b4)32(2a2b4)32=21×32(a2)32(b4)3221×32(a2)32(b4)32=232(a)2×32(b)4×32232(a)2×32(b)42×32=23(a)3(b)623(a)3(b)6=8a3b68a3b6=22a3b6(2a2b4)32=22a3b6Apply the law of exponents for raising a power to a power (am)n=amn.Simplify the expression.Simplify. Apply the rule for rational exponents amn=amn=(an)m m,nεN.Simplify\begin{align*}& (2a^2b^4)^{\frac{3}{2}} && \text{Apply the law of exponents for raising a power to a power} \ \boxed{(a^m)^n=a^{mn}}.\\ & (2a^2b^4)^{\frac{3}{2}}=2^{{\color{red}1 \times \frac{3}{2}}}(a^2)^{{\color{red}\frac{3}{2}}}(b^4)^{{\color{red}\frac{3}{2}}} && \text{Simplify the expression}.\\ & 2^{{\color{red}1 \times \frac{3}{2}}} (a^2)^{{\color{red}\frac{3}{2}}} (b^4)^{{\color{red}\frac{3}{2}}}=2^{{\color{red}\frac{3}{2}}}(a)^{{\color{red}2 \times \frac{3}{2}}}(b)^{{\color{red}4 \times \frac{3}{2}}} && \text{Simplify. Apply the rule for rational exponents} \ \boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m \ m,n \varepsilon N}.\\ & 2^{\color{red}\frac{3}{2}}(a)^{{\color{red}\cancel{2} \times \frac{3}{\cancel{2}}}} (b)^{{\color{red}\overset{2}{\cancel{4}} \times \frac{3}{\cancel{2}}}}=\sqrt{2^{{\color{red}3}}} (a)^{\color{red}3}(b)^{\color{red}6} && \text{Simplify}\\ & \sqrt{2^{\color{red}3}}(a)^{\color{red}3}(b)^{\color{red}6}=\sqrt{{\color{red}8}}a^3b^6\\ & \sqrt{{\color{red}8}}a^3b^6={\color{red}2} \sqrt{{\color{red}2}} a^3b^6\\ & \boxed{(2a^2b^4)^{\frac{3}{2}}=2 \sqrt{2}a^3b^6}\end{align*}

#### Example C

The rational exponents represent the exponent and the index of the base. The numerator is the exponent and the denominator is the index.

a) State the following using radicals:

i) \begin{align*}2^{\frac{3}{8}}\end{align*}

ii) \begin{align*}7^{-\frac{1}{5}}\end{align*}

iii) \begin{align*}3^{\frac{3}{4}}\end{align*}

b) State the following using exponents:

i) \begin{align*}\sqrt[3]{7^2}\end{align*}

ii) \begin{align*}\frac{1}{\left(\sqrt[4]{5}\right)^3}\end{align*}

iii) \begin{align*}\left(\sqrt[5]{a}\right)^2\end{align*}

Solutions:

a) i) \begin{align*}& 2^{\frac{3}{8}}\\ & \boxed{2^{\frac{3}{8}}=\sqrt[{\color{red}8}]{2^{\color{red}3}}=\sqrt[{\color{red}8}]{8}}\end{align*}

ii) \begin{align*}& 7^{-\frac{1}{5}}\\ & 7^{-\frac{1}{5}}=\frac{1}{7^{\frac{1}{5}}}\\ & \boxed{7^{-\frac{1}{5}}=\frac{1}{\sqrt[{\color{red}5}]{7}}}\end{align*}

iii) \begin{align*}& 3^{\frac{3}{4}}\\ & \boxed{3^{\frac{3}{4}}=\sqrt[{\color{red}4}]{3^{\color{red}3}}=\sqrt[{\color{red}4}]{27}}\end{align*}

b) i) \begin{align*}& \sqrt[3]{7^2}\\ & \boxed{\sqrt[3]{7^2}=7^{\color{red}\frac{2}{3}}}\end{align*}

ii) \begin{align*}& \frac{1}{\left(\sqrt[4]{5}\right)^3}\\ & \frac{1}{\left(\sqrt[4]{5}\right)^3}=\frac{1}{5^{\color{red}\frac{3}{4}}}\\ & \frac{1}{5^{\color{red}\frac{3}{4}}}=5^{{\color{red}-\frac{3}{4}}}\end{align*}

iii) \begin{align*}& \left(\sqrt[5]{a}\right)^2\\ & \boxed{\left(\sqrt[5]{a}\right)^2=a^{\color{red}\frac{2}{5}}}\end{align*}

#### Concept Problem Revisited

To determine the value of \begin{align*}4^{\frac{3}{2}}\end{align*}, there are several methods that can be applied.

i) \begin{align*}4^{\frac{3}{2}}\end{align*}

\begin{align*}& 4^{\frac{3}{2}}=(\sqrt{4})^3\end{align*}

‘3’ is the exponent and ‘2’ is the index. Remember the index tells what root to find. The ‘2’ is understood and is never written when the operation is to take the square root of a number or term.

\begin{align*}& (\sqrt{4})^3=({\color{red}2})^3 && \text{The square root of} \ ‘4’ \ \text{is} \ ‘2’.\\ & (2)^3= {\color{red}8}\end{align*}

Therefore, \begin{align*}\boxed{4^{\frac{3}{2}}=8}\end{align*}.

ii) \begin{align*}4^{\frac{3}{2}}\end{align*}

\begin{align*}& 4^{\frac{3}{2}}=\sqrt{4^3} && \text{The first step is to evaluate} \ 4^3.\\ & \sqrt{4^3}=\sqrt{{\color{red}64}} && \text{Take the square root of} \ 64.\\ & \sqrt{64}={\color{red}8}\end{align*}

Therefore, \begin{align*}\boxed{4^{\frac{3}{2}}=8}\end{align*}.

iii) \begin{align*}4^{\frac{3}{2}}\end{align*}

\begin{align*}& 4^{\frac{3}{2}}=\left(4^{\color{red}\frac{1}{2}}\right)^{\color{red}3} && \text{Write the power as} \ \left(a^{\frac{1}{n}}\right)^m.\\ & \left(4^{\color{red}\frac{1}{2}}\right)^{\color{red}3}=\left({\color{red}\sqrt{4}}\right)^{3} && 4^{\frac{1}{2}} \ \text{means to take the square root of} \ 4.\\ & \left(\sqrt{4}\right)^3={\color{red}2}^3 && \text{Evaluate} \ 2^3.\\ & 2^3={\color{red}8}\end{align*}

Therefore, \begin{align*}\boxed{4^{\frac{3}{2}}=8}\end{align*}.

iv) \begin{align*}& 4^{\frac{3}{2}} && \text{Write} \ 4 \ \text{as a power of} \ ‘2’.\\ & 4^{\frac{3}{2}}=(2^2)^\frac{3}{2} && \text{Raise the power to a power by multiplying the exponents.}\\ & (2^2)^{\frac{3}{2}}=2^{{\color{red}2 \times \frac{3}{2}}} && \text{Multiply the exponents.}\\ & 2^{\cancel{2} \times \frac{3}{\cancel{2}}}=2^{\color{red}3} && \text{Evaluate} \ 2^3.\\ & 2^3={\color{red}8}\end{align*}

Therefore, \begin{align*}\boxed{4^{\frac{3}{2}}=8}\end{align*}.

### Vocabulary

Base
In an algebraic expression, the base is the variable, number, product or quotient, to which the exponent refers. Some examples are: In the expression \begin{align*}2^5\end{align*}, ‘2’ is the base. In the expression \begin{align*}(-3y)^4\end{align*}, ‘\begin{align*}-3y\end{align*}’ is the base.
Exponent
In an algebraic expression, the exponent is the number to the upper right of the base that tells how many times to multiply the base times itself. Some examples are:
In the expression \begin{align*}2^5\end{align*}, ‘5’ is the exponent. It means to multiply 2 times itself 5 times as shown here: \begin{align*}2^5=2 \times 2 \times 2 \times 2 \times 2\end{align*}
In the expression \begin{align*}(-3y)^4\end{align*}, ‘4’ is the exponent. It means to multiply \begin{align*}-3y\end{align*} times itself 4 times as shown here: \begin{align*}(-3y)^4=-3y \times -3y \times -3y \times -3y\end{align*}.
Laws of Exponents
The laws of exponents are the algebra rules and formulas that tell us the operation to perform on the exponents when dealing with exponential expressions.

### Guided Practice

1. Use the laws of exponents to evaluate the following: \begin{align*}9^{\frac{3}{2}} \div 36^{-\frac{1}{2}}\end{align*}

2. Simplify the following using the laws of exponents. \begin{align*}(20a^2b^3c^{-1})^{\frac{3}{2}}\end{align*}

3. Use the laws of exponents to evaluate the following: \begin{align*}\frac{64^{\frac{2}{3}}}{216^{-\frac{1}{3}}}\end{align*}

1. \begin{align*}9^{\frac{3}{2}} \div 36^{-\frac{1}{2}}\end{align*}

Apply the law of exponents for rational exponents \begin{align*}\boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m \ m, n \varepsilon N} to 9^\left\{\frac\left\{3\right\}\left\{2\right\}\right\}\end\left\{align*\right\}.
Apply the law of exponents for negative exponents \begin{align*}\boxed{a^{-m}=\frac{1}{a^m}} to 36^\left\{-\frac\left\{1\right\}\left\{2\right\}\right\}\end\left\{align*\right\}.
\begin{align*}& 9^{\frac{3}{2}} \div 36^{-\frac{1}{2}}=\left({\color{red}\sqrt{9}}\right)^{\color{red}3} \div \frac{1}{36^{\color{red}\frac{1}{2}}} && \text{Find the square root of} \ 9. \ \text{Apply} \ \boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m \ m,n \varepsilon N} \ 36^{\frac{1}{2}}.\\ & \left(\sqrt{9}\right)^3 \div \frac{1}{36^{\frac{1}{2}}}=({\color{red}3})^3 \div \frac{1}{{\color{red}\sqrt{36}}} && \text{Simplify}\\ & (3)^3 \div \frac{1}{\sqrt{36}}={\color{red}27} \div \frac{1}{\sqrt{6}} && \text{Perform the indicated operation of division.}\\ & 27 \div \frac{1}{6}=27 \times \frac{6}{1}={\color{red}162}\\ & \boxed{9^{\frac{3}{2}} \div 36^{-\frac{1}{2}}=162}\end{align*}

2.

\begin{align*}& (20a^2b^3c^{-1})^{\frac{3}{2}} && \text{Apply the law of exponents} \ \boxed{(ab)^n=a^nb^n}.\\ & (20a^2b^3c^{-1})^{\frac{3}{2}}=20^{{\color{red}1 \times \frac{3}{2}}} (a)^{{\color{red}2 \times \frac{3}{2}}} (b)^{{\color{red}3 \times \frac{3}{2}}} (c)^{{\color{red}-1 \times \frac{3}{2}}} && \text{Simplify the exponents}.\\ & 20^{1 \times \frac{3}{2}} (a)^{2 \times \frac{3}{2}} (b)^{3 \times \frac{3}{2}} (c)^{-1 \times \frac{3}{2}}=20^{\color{red}\frac{3}{2}} (a)^{\color{red}3} (b)^{\color{red}\frac{9}{2}} (c)^{\color{red}-\frac{3}{2}}\end{align*}
Apply the law of exponents for rational exponents \begin{align*}\boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m m, n \varepsilon N}\end{align*} to \begin{align*}20^{\frac{3}{2}}\end{align*} and \begin{align*}b^{\frac{9}{2}}\end{align*}. To \begin{align*}c^{-\frac{3}{2}}\end{align*}, apply the law for negative exponents \begin{align*}\boxed{a^{-m}=\frac{1}{a^m}}\end{align*} and then the law \begin{align*}\boxed{a^{\frac{m}{n}}=\sqrt[n]{a^m}=\left(\sqrt[n]{a}\right)^m \ m,n \varepsilon N}\end{align*} for rational exponents.
\begin{align*}& 20^{\frac{3}{2}} (a)^3 (b)^{\frac{9}{2}} (c)^{-\frac{3}{2}}=\left({\color{red}\sqrt{20}}\right)^{\color{red}3} (a)^3 {\color{red}\sqrt{b^9}} \left({\color{red}\frac{1}{c^{\frac{3}{2}}}}\right) && \text{Simplify}\\ & 20^{\frac{3}{2}} (a)^3 (b)^{\frac{9}{2}} (c)^{-\frac{3}{2}}=\left({\color{red} 2 \sqrt{5}}\right)^{\color{red}3} {\color{red}(a^3) \left ( b^4 \sqrt{b} \right )} \left({\color{red}\frac{1}{\sqrt{c^3}}}\right) && \text{Simplify}\\ & \left(2 \sqrt{5}\right)^3 (a^3) \left ( b^4 \sqrt{b}\right) \left(\frac{1}{\sqrt{c^3}}\right)={\color{red}8\sqrt{125}} a^3b^4 \sqrt{b} \frac{1}{{\color{red}c \sqrt{c}}}\\ & 8\sqrt{125}a^3b^4 \sqrt{b}\frac{1}{c\sqrt{c}}={\color{red}40\sqrt{5}} a^3b^4 \sqrt{b} \left({\color{red}c \sqrt{c}}\right)^{{\color{red}-1}} && \text{Simplify}\\ & \boxed{(20 a^2b^3c^{-1})^{\frac{3}{2}}=40 \sqrt{5}a^3b^4 \sqrt{b} \left(c\sqrt{c}\right)^{-1}}\end{align*}

3. \begin{align*}\frac{64^{\frac{2}{3}}}{216^{-\frac{1}{3}}}\end{align*}.

\begin{align*}& \mathbf{Numerator} && \mathbf{Denominator}\\ & 64^{\frac{2}{3}} && 216^{-\frac{1}{3}}\\ & 64^{\frac{2}{3}}=\left(\sqrt[3]{64}\right)^2 && 216^{-\frac{1}{3}}=\frac{1}{216^{\frac{1}{3}}}\\ & \left(\sqrt[3]{64}\right)^2=(4)^2 && 216^{-\frac{1}{3}}=\frac{1}{\sqrt[3]{216}}\\ & (4)^2=16 && \frac{1}{\sqrt[3]{216}}=\frac{1}{6}\end{align*}
Numerator divided by denominator:
\begin{align*}& 16 \div \frac{1}{6}\\ & 16 \times \frac{6}{1}=96\\ & \boxed{\frac{64^{\frac{2}{3}}}{216^{-\frac{1}{3}}}=96}\end{align*}

### Practice

Express each of the following as a radical and if possible, simplify.

1. \begin{align*}x^{\frac{1}{2}}\end{align*}
2. \begin{align*}5^{\frac{3}{4}}\end{align*}
3. \begin{align*}2^{\frac{3}{2}}\end{align*}
4. \begin{align*}2^{-\frac{1}{2}}\end{align*}
5. \begin{align*}9^{-\frac{1}{5}}\end{align*}

Express each of the following using exponents:

1. \begin{align*}\sqrt{26}\end{align*}
2. \begin{align*}\sqrt[3]{5^2}\end{align*}
3. \begin{align*}\left(\sqrt[6]{a}\right)^5\end{align*}
4. \begin{align*}\sqrt[4]{m}\end{align*}
5. \begin{align*}\left(\sqrt[3]{7}\right)^2\end{align*}

Evaluate each of the following using the laws of exponents:

1. \begin{align*}3^{\frac{2}{5}} \times 3^{\frac{3}{5}}\end{align*}
2. \begin{align*}(6^{0.4})^5\end{align*}
3. \begin{align*}2^{\frac{1}{7}} \times 4^{\frac{3}{7}}\end{align*}
4. \begin{align*}\left(\frac{64}{125}\right)^{-\frac{1}{2}}\end{align*}
5. \begin{align*}(81^{-1})^{-\frac{1}{4}}\end{align*}

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### Vocabulary Language: English Spanish

roots as fractional exponents

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

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