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Found in: Page 266

### Precalculus Mathematics for Calculus

Book edition 7th Edition
Author(s) James Stewart, Lothar Redlin, Saleem Watson
Pages 948 pages
ISBN 9781337067508

# Radicals and Exponents Evaluate each expression.a. ${\mathbf{3}}^{\mathbf{8}}\mathbf{\cdot }{\mathbf{3}}^{\mathbf{5}}$b. $\frac{{\mathbf{10}}^{\mathbf{7}}}{{\mathbf{10}}^{\mathbf{4}}}$c. ${\mathbf{\left(}{\mathbf{3}}^{\mathbf{5}}\mathbf{\right)}}^{\mathbf{4}}$

a) The value of${3}^{8}\cdot {3}^{5}$is $\mathbf{1594323}$.

b) The value of $\frac{{10}^{7}}{{10}^{4}}$ islocalid="1648708601105" $\mathbf{1000}$.

c) The value oflocalid="1648707916158" ${\left({3}^{5}\right)}^{4}$is$\mathbf{3486784401}$.

See the step by step solution

## Part a Step 1. Given information.

The given expression is ${3}^{8}\cdot {3}^{5}$.

## Part a Step 2. Write the concept.

According to the property of exponent:

${a}^{m}\cdot {a}^{n}={a}^{m+n}$

The exponential expression “$a$ to the power $n$” is defined as:

${a}^{n}=a×a×a×\cdots ×a\text{\hspace{0.17em}}\left(n\text{\hspace{0.17em}times}\right)$

## Part a Step 3. Determine the value of the expression.

The given expression can be written as:

$\begin{array}{l}{3}^{8}\cdot {3}^{5}={3}^{8+5}\\ ={3}^{13}\\ =3×3×3×3×3×3×3×3×3×3×3×3×3\\ =1594323\end{array}$

Thus, the value of ${3}^{8}\cdot {3}^{5}$ is $\mathbf{1594323}$.

## Part b Step 1. Given information.

The given expression is $\frac{{10}^{7}}{{10}^{4}}$.

## Part b Step 2. Write the concept.

According to the property of exponent:

$\frac{{a}^{m}}{{a}^{n}}={a}^{m-n}$

The exponential expression “$a$ to the power $n$” is defined as:

${a}^{n}=a×a×a×\cdots ×a\text{\hspace{0.17em}}\left(n\text{\hspace{0.17em}times}\right)$

## Part b Step 3. Determine the value of the expression.

The given expression can be written as:

$\begin{array}{l}\frac{{10}^{7}}{{10}^{4}}={10}^{7-4}\\ ={10}^{3}\\ =10×10×10\\ =1000\end{array}$

Thus, the value of $\frac{{10}^{7}}{{10}^{4}}$ is $\mathbf{1000}$.

## Part c Step 1. Given information.

The given expression is${\left({3}^{5}\right)}^{4}$.

## Part c Step 2. Write the concept.

According to the property of exponent:

${\left({a}^{m}\right)}^{n}={a}^{mn}$

The exponential expression “$a$ to the power $n$” is defined as:

${a}^{n}=a×a×a×\cdots ×a\text{\hspace{0.17em}}\left(n\text{\hspace{0.17em}times}\right)$

## Part c Step 3. Determine the value of the expression.

The given expression can be written as:

$\begin{array}{l}{\left({3}^{5}\right)}^{4}={3}^{5×4}\\ ={3}^{20}\\ =3×3×3×3×3×3×3×3×3×3×3×3×3×3×3×3×3×3×3×3\\ =3486784401\end{array}$

Thus, the value of ${\left({3}^{5}\right)}^{4}$ is $\mathbf{3486784401}$.