Chapter 7.2, Problem 4BGP

To determine

To Simplify: The Given expression

$\frac{32a^{-8}{b}^3{c}^{-4}}{4a^3{b}^5{c}^{-2}}$

Answer to Problem 4BGP

The simplified expression is $\frac{8}{a^{11}{b}^2{c}^2}$ .

Explanation of Solution

Given information:

$\frac{32a^{-8}{b}^3{c}^{-4}}{4a^3{b}^5{c}^{-2}}$

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:

  $\frac{32a^{-8}{b}^3{c}^{-4}}{4a^3{b}^5{c}^{-2}}$

(Group the powers with the same base)

  $=\frac{32}{4}\cdot \left(\frac{a^{-8}}{a^3}\right)\left(\frac{b^3}{b^5}\right)\left(\frac{c^{-4}}{c^{-2}}\right)$

Using Quotient of Powers

  $=\frac{32}{4}\cdot a^{-8-3}{b}^{3-5}{c}^{-4-\left(-2\right)}$

Simplify.

  $\begin{array}{l}=8\cdot a^{-11}{b}^{-2}{c}^{-4+2}\\ =8\cdot a^{-11}{b}^{-2}{c}^{-2}\end{array}$

Using Negative Exponent Property

  $=\frac{8}{a^{11}{b}^2{c}^2}$