Chapter 1.3, Problem 7E

To determine

To find: The expression ${\left(\frac{1}{8}\right)}^{2x}$ in terms of the expression with base $2$.

Answer to Problem 7E

The expression ${\left(\frac{1}{8}\right)}^{2x}$ is written as ${\left(2\right)}^{-6x}$ in terms of expression with base $2$.

Explanation of Solution

Given information:

The expression is ${\left(\frac{1}{8}\right)}^{2x}$ and the indicated base is $2$.

Calculation:

Simplify the expression in terms of $2$.

  $\begin{array}{c}{\left(\frac{1}{8}\right)}^{2x}={\left(\frac{1}{2^3}\right)}^{2x}\\ ={\left(2^{-3}\right)}^{2x}\left[\therefore {\left(x^a\right)}^b=x^{ab}\right]\\ =2^{-6x}\end{array}$

Therefore, the expression ${\left(\frac{1}{8}\right)}^{2x}$ is written as ${\left(2\right)}^{-6x}$ in terms of expression with base $2$.