Chapter 7.4, Problem 61PPE

To determine

To simplify: the expression ${\left(\frac{2m^4}{m^2}\right)}^{-4}$

Answer to Problem 61PPE

The required answer is: $\frac{0.0625}{m^8}$

Explanation of Solution

Given information:

The given expression: ${\left(\frac{2m^4}{m^2}\right)}^{-4}$

Calculation:

  ${\left(\frac{2m^4}{m^2}\right)}^{-4}=\frac{1}{{\left(\frac{2m^4}{m^2}\right)}^4}$

(Rewrite using positive exponents)

  $=\frac{1}{\frac{{(2m^4)}^4}{{(m^2)}^4}}$

(Raise the quotient to the fourth power)

  $=\frac{1}{\frac{2^4{m}^{16}}{m^8}}$

(Multiply the exponents)

  $=1\cdot \frac{m^8}{2^4{m}^{16}}$

(Multiply by the reciprocal of $\frac{2^4{m}^{16}}{m^8}$ which is $\frac{m^8}{2^4{m}^{16}}$ )

  $=\frac{1}{2^4}{m}^{8-16}$

(Subtract exponents when dividing powers with the same base)

  $\begin{array}{l}=\frac{1}{16}{m}^{-8}\\ =\frac{1}{16}\cdot \frac{1}{m^8}\end{array}$

(Rewrite using positive exponents)

  $=\frac{0.0625}{m^8}$

(Simplify)