Chapter 7.2, Problem 40PPS

To determine

To Simplify: The Given expression

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

Answer to Problem 40PPS

The simplified expression is $a^4{b}^{16}{c}^4$ .

Explanation of Solution

Given information:

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

Concept Used:

Power of a quotient: It states that in order to find the power of a quotient, find the power of the numerator and that of the denominator.

Example: let a and b be any non-zero number and m be any integer.

  ${\left(\frac{a}{b}\right)}^m=\frac{a^m}{b^m}$

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.

  $\frac{a^m}{a^n}=a^{m-n}$

Calculation:

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

Using Power of Quotient

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

Simplify

  $=\frac{a^{-4}{b}^8{c}^{10}}{a^{-8}{b}^{-8}{c}^6}$

(Group the powers with the same base)

  $=\left(\frac{a^{-4}}{a^{-8}}\right)\left(\frac{b^8}{b^{-8}}\right)\left(\frac{c^{10}}{c^6}\right)$

Using Quotient of Powers

  $=a^{-4-\left(-8\right)}{b}^{8-\left(-8\right)}{c}^{10-6}$

Simplify

  $=a^4{b}^{16}{c}^4$