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Q21.

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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. ${\mathbf{5}}^{\mathbf{3}}\mathbf{\cdot }\mathbf{5}$b. ${\mathbf{5}}^{\mathbf{4}}\mathbf{\cdot }{\mathbf{5}}^{\mathbf{-}\mathbf{2}}$c. ${\mathbf{\left(}{\mathbf{2}}^{\mathbf{2}}\mathbf{\right)}}^{\mathbf{3}}$

a) The value of ${5}^{3}\cdot 5$ is 625.

b) The value of ${5}^{4}\cdot {5}^{-2}$ is 25.

c) The value of ${\left({2}^{2}\right)}^{3}$ is 64.

See the step by step solution

## Part a Step 1. Given information.

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

## 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:

$\begin{array}{l}{5}^{3}\cdot 5={5}^{3+1}\\ ={5}^{4}\\ =5×5×5×5\\ =625\end{array}$

Thus, the value of ${5}^{3}\cdot 5$ is 625.

## Part b Step 1. Given information.

The given expression is ${5}^{4}\cdot {5}^{-2}$.

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

The given expression can be written as:

$\begin{array}{l}{5}^{4}\cdot {5}^{-2}={5}^{4+\left(-2\right)}\\ ={5}^{4-2}\\ ={5}^{2}\\ =5×5\\ =25\end{array}$

Thus, the value of ${5}^{4}\cdot {5}^{-2}$ is 25.

## Part c Step 1. Given information.

The given expression is ${\left({2}^{2}\right)}^{3}$.

## Part c Step 2. Write the concept.

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:

$\begin{array}{l}{\left({2}^{2}\right)}^{3}={\left(4\right)}^{3}\\ =\left(4\right)×\left(4\right)×\left(4\right)\\ =64\end{array}$

Thus, the value of ${\left({2}^{2}\right)}^{3}$ is 64.