Chapter 7.4, Problem 77PPE

To determine

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

Answer to Problem 77PPE

The required answer is: $\frac{0.0039y^6}{x^2}$

Explanation of Solution

Given information:

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

Calculation:

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

(Rewrite using positive exponents)

  $=\frac{1}{\frac{{\left(4^2{x}^{2}y\right)}^2}{{\left(x{y}^4\right)}^2}}$

(Raise the quotient to the second power)

  $=\frac{1}{\frac{4^4{x}^4{y}^2}{x^2{y}^8}}$

(Multiply the exponents)

  $=1\cdot \frac{x^2{y}^8}{4^4{x}^4{y}^2}$

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

  $=\frac{1}{4^4}{x}^{2-4}{y}^{8-2}$

(Subtract exponents when dividing powers with the same base)

  $\begin{array}{l}=\frac{1}{256}{x}^{-2}{y}^6\\ =\frac{1}{256}\cdot \frac{1}{x^2}{y}^6\end{array}$

(Rewrite using positive exponents)

  $=\frac{0.0039y^6}{x^2}$

(Simplify)