Chapter 7.2, Problem 16CYU

To determine

To Simplify: The Given expression

$\frac{-8x^2{y}^8{z}^{-5}}{12x^4{y}^{-7}{z}^7}$

Answer to Problem 16CYU

The simplified expression is $-\frac{2y^{15}}{3x^2{z}^{12}}$ .

Explanation of Solution

Given information:

$\frac{-8x^2{y}^8{z}^{-5}}{12x^4{y}^{-7}{z}^7}$

Concept Used:

Quotient of powers: It states that to divide two powers with same base, subtract the exponent.

For example, let a be any non-zero number and m and n be any integer.

  $\frac{a^m}{a^n}=a^{m-n}$

Negative Exponent Property: $a^{-n}$ is the reciprocal of $a^n$ , where a is a non-zero number and n is an integer.

  $a^{-n}=\frac{1}{a^n}$

Calculation:

  $\frac{-8x^2{y}^8{z}^{-5}}{12x^4{y}^{-7}{z}^7}$

(Group the powers with the same base)

  $=\left(\frac{-8}{12}\right)\left(\frac{x^2}{x^4}\right)\left(\frac{y^8}{y^{-7}}\right)\left(\frac{z^{-5}}{z^7}\right)$

Using Quotient of Powers

  $=\left(\frac{-8}{12}\right)x^{2-4}{y}^{8-\left(-7\right)}{z}^{-5-7}$

Simplify.

  $\begin{array}{l}=-\frac{2}{3}{x}^{-2}{y}^{8+7}{z}^{-12}\\ =-\frac{2}{3}{x}^{-2}{y}^{15}{z}^{-12}\end{array}$

Applying Negative Exponent Property

  $\begin{array}{l}=-\frac{2}{3}\cdot \frac{1}{x^2}{y}^{15}\frac{1}{z^{12}}\\ =-\frac{2y^{15}}{3x^2{z}^{12}}\end{array}$