Chapter 7.9, Problem 58WE

To determine

To calculate: The simplified form of the expression ${\left(\frac{y^5}{y^{-2}}\right)}^{-3}$ .

Answer to Problem 58WE

The simplified form of the expression ${\left(\frac{y^5}{y^{-2}}\right)}^{-3}$ is $\frac{1}{y^{21}}$ .

Explanation of Solution

Given information:

The expression ${\left(\frac{y^5}{y^{-2}}\right)}^{-3}$ .

Formula used:

Law of exponent for quotient of powers is $a^m÷a^n=a^{m-n}$ where a is a nonzero integers and n and m are positive integers.

Law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

The expression $a^{-n}$ is expressed as $\frac{1}{a^n}$ where a is a nonzero real number and n is a positive integer.

Calculation:

Consider the expression ${\left(\frac{y^5}{y^{-2}}\right)}^{-3}$ .

Recall that law of exponent for quotient of powers is $a^m÷a^n=a^{m-n}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}{\left(\frac{y^5}{y^{-2}}\right)}^{-3}={\left(y^{5-\left(-2\right)}\right)}^{-3}\\ ={\left(y^7\right)}^{-3}\end{array}$

Recall that law of exponent for power of power is ${\left(a^m\right)}^n=a^{mn}$ where a is a nonzero integers and n and m are positive integers.

Apply it,

  $\begin{array}{c}{\left(\frac{y^5}{y^{-2}}\right)}^{-3}={\left(y^{5-\left(-2\right)}\right)}^{-3}\\ ={\left(y^7\right)}^{-3}\\ =y^{7\left(-3\right)}\\ =y^{-21}\end{array}$

Recall that expression $a^{-n}$ is expressed as

  $\frac{1}{a^n}$ where a is a nonzero real number and n is a positive integer.

Apply it,

  $\begin{array}{c}{\left(\frac{y^5}{y^{-2}}\right)}^{-3}=y^{7\left(-3\right)}\\ =y^{-21}\\ =\frac{1}{y^{21}}\end{array}$

Thus, the simplified form of the expression ${\left(\frac{y^5}{y^{-2}}\right)}^{-3}$ is $\frac{1}{y^{21}}$ .