Chapter 7, Problem 45CR

To determine

To simplify: The expression ${\left(\frac{3c^3}{e^5}\right)}^{-4}$

Answer to Problem 45CR

  $\frac{e^{20}}{81c^{12}}$

Explanation of Solution

Given information: The expression ${\left(\frac{3c^3}{e^5}\right)}^{-4}$

Calculation:

  $\begin{array}{l}{\left(\frac{3c^3}{e^5}\right)}^{-4}\\ =\frac{{\left(3c^3\right)}^{-4}}{{\left(e^5\right)}^{-4}}[\because {\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}]\\ $ =\frac{3^{-4}{(c^3)}^{-4}}{{(e^5)}^{-4}}[\because {(ab)}^m=a^m{b}^m]$$ =\frac{3^{-4}{c}^{3\cdot (-4)}}{e^{5\cdot (-4)}}[\because {(a^m)}^n=a^{mn}]$=\frac{3^{-4}{c}^{-12}}{e^{-20}}\\ =\frac{e^{20}}{3^4{c}^{12}}[\because a^{-m}=\frac{1}{a^m}]\\ =\frac{e^{20}}{81c^{12}}\end{array}$

Thus, ${\left(\frac{3c^3}{e^5}\right)}^{-4}=\frac{e^{20}}{81c^{12}}$