Chapter 7.9, Problem 21WE

To determine

To calculate: The simplified form of the expression ${\left(\frac{3^{-1}}{3}\right)}^2$ .

Answer to Problem 21WE

The simplified form of the expression ${\left(\frac{3^{-1}}{3}\right)}^2$ is $\frac{1}{81}$ .

Explanation of Solution

Given information:

The expression ${\left(\frac{3^{-1}}{3}\right)}^2$ .

Formula used:

Law of exponent for power of quotient is ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$ where a andb are nonzero integers and n is a positive integer.

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

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.

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.

Calculation:

Consider the expression ${\left(\frac{3^{-1}}{3}\right)}^2$ .

Recall that law of exponent for power of quotient is ${\left(\frac{a}{b}\right)}^n=\frac{a^n}{b^n}$ where a andb are nonzero integers and n is a positive integer.

Apply it,

  ${\left(\frac{3^{-1}}{3}\right)}^2=\frac{{\left(3^{-1}\right)}^2}{3^2}$

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,

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

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,

  $\begin{array}{c}{\left(\frac{3^{-1}}{3}\right)}^2=\frac{{\left(3^{-1}\right)}^2}{3^2}\\ =\frac{3^{-2}}{3^2}\\ =3^{-2-2}\\ =3^{-4}\end{array}$

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,

  $\begin{array}{c}{\left(\frac{3^{-1}}{3}\right)}^2=3^{-4}\\ =\frac{1}{3^4}\\ =\frac{1}{81}\end{array}$

Thus, the simplified form of the expression ${\left(\frac{3^{-1}}{3}\right)}^2$ is $\frac{1}{81}$ .