Chapter 11, Problem 8GR

To determine

Simplify: $\frac{8x^{-3}{y}^2{z}^4}{5x^{3}y{z}^{-2}}$

Answer to Problem 8GR

  $\frac{8x^{-3}{y}^2{z}^4}{5x^{3}y{z}^{-2}}=\frac{8y{z}^6}{5x^6}$

Explanation of Solution

Given:

Simplify: $\frac{8x^{-3}{y}^2{z}^4}{5x^{3}y{z}^{-2}}$

Concept Used:

Use the exponent rules:

  $\text{Negative}\text{exponent}\text{rule}:a^{-n}=\frac{1}{a^n}or\frac{1}{a^{-n}}=a^n$

  $\text{Product}\text{rule}=a^m·a^n=a^{m+n}$

  $\text{Quotient}\text{rule}:\frac{a^{ m}}{a^n}=a^{m-n}$

Calculation:

Use the exponent rules:

  $\begin{array}{l}\frac{8x^{-3}{y}^2{z}^4}{5x^3{y}^1{z}^{-2}}=\frac{8·y^2·z^4·z^2}{5x^3·x^3·y^1}[a^{-n}=\frac{1}{a^n}\text{or}\frac{1}{a^{-n}}=a^n]\\ \frac{8·y^{2-1}·z^{4+2}}{5·x^{3+3}}=\frac{8·y·z^6}{5·x^6}=\frac{8y{z}^6}{5x^6}[a^m·a^n=a^{m+n}\text{and}\frac{a^{ m}}{a^n}=a^{m-n}]\end{array}$

Thus, $\frac{8x^{-3}{y}^2{z}^4}{5x^{3}y{z}^{-2}}=\frac{8y{z}^6}{5x^6}$