Chapter 4, Problem 1RE

Applying Properties of Exponents In Exercises 1 and 2, use the properties of exponents to simplify each expression.

(a) $\left(4^5\right)\left(4^2\right)$

(b) ${\left(7^3\right)}^3$

(c) $2^{-4}$

(d) $\frac{9^8}{9^6}$

To determine

To calculate: The value of the expression $\left(4^5\right)\left(4^2\right)$ using the properties of exponents.

Answer to Problem 1RE

Solution:

The value of the expression $\left(4^5\right)\left(4^2\right)$ is $16384$.

Explanation of Solution

Given Information:

The provided expression is $\left(4^5\right)\left(4^2\right)$.

Formula used:

Property of exponents:

For any real number $x$ and $y$,

$a^x{a}^y=a^{x+y}$

Calculation:

Consider the provided expression, $\left(4^5\right)\left(4^2\right)$

Here, the base of the expression is the same for two exponents that is $4$.

Evaluate the value of expression by applying product rule.

$\begin{array}{c}\left(4^5\right)\left(4^2\right)=4^{5+2}\\ =4^7\\ =16384\end{array}$

Thus, the value of the expression $\left(4^5\right)\left(4^2\right)$ is $16384$.

To determine

To calculate: The value of the expression ${\left(7^3\right)}^3$ using the properties of exponents.

Answer to Problem 1RE

Solution:

The value of the expression ${\left(7^3\right)}^3$ is $40,353,607$.

Explanation of Solution

Given Information:

The provided expression is ${\left(7^3\right)}^3$.

Formula used:

If the expression is in the form of ${\left(x^a\right)}^b$, then the power rule of exponents is given by,

${\left(x^a\right)}^b={\left(x\right)}^{ab}$

Calculation:

Consider the provided expression, ${\left(7^3\right)}^3$

Evaluate the value of expression by applying power rule.

$\begin{array}{c}{\left(7^3\right)}^3=7^{3\cdot 3}\\ =7^9\\ =40353607\end{array}$

Thus, the value of the expression ${\left(7^3\right)}^3$ is $40,353,607$.

To determine

To calculate: The value of the expression ${\left(2\right)}^{-4}$ using the properties of exponents.

Answer to Problem 1RE

Solution:

The value of the expression ${\left(2\right)}^{-4}$ is $0.0625$.

Explanation of Solution

Given Information:

The provided expression is ${\left(2\right)}^{-4}$.

Formula used:

If the expression is in the form of $\frac{1}{a^{-x}}$, then the negative rule of exponents defines that the negative power in the denominator moved to the numerator and become positive exponent which is given by,

$\frac{1}{a^{-x}}=a^x$

Calculation:

Consider the provided expression, ${\left(2\right)}^{-4}$

Evaluate the value of expression by applying negative exponent rule.

$\begin{array}{c}{\left(2\right)}^{-4}=\frac{1}{2^4}\\ =\frac{1}{16}\\ =0.0625\end{array}$

Thus, the value of the expression ${\left(2\right)}^{-4}$ is $0.0625$.

To determine

To calculate: The value of the expression $\frac{9^8}{9^6}$ using the properties of exponents.

Answer to Problem 1RE

Solution:

The value of the expression $\frac{9^8}{9^6}$ is $81$.

Explanation of Solution

Given Information:

The provided expression is $\frac{9^8}{9^6}$.

Formula used:

If the expression is in the form of $\frac{a^x}{a^y}$, then the quotient rule of exponents defines that if the two exponents are divided with the same base, then the powers are subtracted, and is given by.

$\frac{a^x}{a^y}=a^{x-y}$

Calculation:

Consider the provided expression, $\frac{9^8}{9^6}$

Evaluate the value of expression by applying quotient rule.

$\begin{array}{c}\frac{9^8}{9^6}=9^{8-6}\\ =9^2\\ =81\end{array}$

Thus, the value of the expression is $81$.