Chapter R.19, Problem 1DE

To determine

To calculate: The simplified form of the expression $8^{-2}\cdot 8^{-4}$.

Answer to Problem 1DE

Solution:

The expression $8^{-2}\cdot 8^{-4}$ is simplified as $\underset{\_}{\frac{1}{8^6}}$.

Explanation of Solution

Given information:

The provided expression is $8^{-2}\cdot 8^{-4}$.

Formula used:

For any real number a and b and any integer m and n,

Product rule is as below:

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

The negative exponent rule:

For any non-zero real number a and any integer n,

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

Calculation:

Consider the expression:

$8^{-2}\cdot 8^{-4}$

Use the product rule, and simplify:

$\begin{array}{c}{8}^{-2}\cdot 8^{-4}=8^{-2+\left(-4\right)}\\ =8^{-6}\end{array}$

Use the negative exponent rule as below:

$\begin{array}{c}{8}^{-2}\cdot 8^{-4}=8^{-6}\\ =\frac{1}{8^6}\end{array}$

Hence, the expression $8^{-2}\cdot 8^{-4}$ is simplified as $\frac{1}{8^6}$.