Chapter 7.9, Problem 48WE

To determine

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

Answer to Problem 48WE

The simplified form of the expression $3x^{-2}\cdot {\left(3x^2\right)}^{-1}$ is $\frac{1}{x^4}$ .

Explanation of Solution

Given information:

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

Formula used:

Law of exponent for product of power is $a^m\cdot a^n=a^{m+n}$ where a andb are nonzero integers and n and m are positive integers.

Calculation:

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

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

Apply it,

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

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,

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

Recall that law of exponent for product of power is $a^m\cdot a^n=a^{m+n}$ where a andb are nonzero integers and n and m are positive integers.

Apply it,

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

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,

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

Thus, the simplified form of the expression $3x^{-2}\cdot {\left(3x^2\right)}^{-1}$ is $\frac{1}{x^4}$ .