Chapter 1.3, Problem 8E

To determine

To find: The expression ${\left(\frac{1}{27}\right)}^x$ in terms of the expression with base $3$.

Answer to Problem 8E

The expression ${\left(\frac{1}{27}\right)}^x$ is written as ${\left(3\right)}^{-3x}$ in terms of expression with base $3$.

Explanation of Solution

Given information:

The expression is ${\left(\frac{1}{27}\right)}^x$ and the indicated base is $3$.

Calculation:

Simplify the expression in terms of $3$.

  $\begin{array}{c}{\left(\frac{1}{27}\right)}^x={\left(\frac{1}{3^3}\right)}^x\\ ={\left(3^{-3}\right)}^x\left[\therefore {\left(x^a\right)}^b=x^{ab}\right]\\ =3^{-3x}\end{array}$

Therefore, the expression ${\left(\frac{1}{27}\right)}^x$ is written as ${\left(3\right)}^{-3x}$ in terms of expression with base $3$.