Chapter 7.4, Problem 4P

To determine

To write: the simplified form of expression ${\left(\frac{a}{5b}\right)}^{-2}$

Answer to Problem 4P

The required answer is: $\frac{25b^2}{a^2}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{a}{5b}\right)}^{-2}$

Calculation:

  ${\left(\frac{a}{5b}\right)}^{-2}=\frac{1}{{\left(\frac{a}{5b}\right)}^2}$

(Rewrite using positive exponents)

  $=\frac{1}{\frac{a^2}{{(5b)}^2}}$

(Raise the quotient to the second power)

  $=\frac{1}{\frac{a^2}{5^2{b}^2}}$

(Multiply the exponents)

  $=1\cdot \frac{5^2{b}^2}{a^2}$

(Multiply with the reciprocal of $\frac{a^2}{5^2{b}^2}$ which is $\frac{5^2{b}^2}{a^2}$ )

  $\begin{array}{l}=\frac{5^2{b}^2}{a^2}\\ =\frac{25b^2}{a^2}\end{array}$

(Simplify)