Chapter P.1, Problem 49E

In Exercises 47−52, simplify the expression. Assume that the variables in the denominators are nonzero.

49. ${\left(\frac{4}{x^2}\right)}^2$

Solution

To simplify the expression ${\left(\frac{4}{x^2}\right)}^2$ .

  $\frac{16}{x^4}$

Given:

The expression, ${\left(\frac{4}{x^2}\right)}^2$ .

Concept Used:

Exponent Properties:

• Product property: $a^m\cdot a^n=a^{m+n}$
• Quotient property: $\frac{a^m}{a^n}=a^{m-n}$
• Power of a power property: ${\left(a^m\right)}^n=a^{m\cdot n}$
• Power to product property: ${\left(ab\right)}^n=a^n\cdot b^n$
• Negative exponent property: $a^{-n}=\frac{1}{a^n}$

Calculation:

In order to solve the given expression ${\left(\frac{4}{x^2}\right)}^2$ , use the exponent properties and simplify further as shown below,

  $\begin{array}{c}{\left(\frac{4}{x^2}\right)}^2=\frac{4^2}{{\left(x^2\right)}^2}\\ =\frac{16}{x^{2\cdot 2}}\\ =\frac{16}{x^4}\end{array}$

Therefore, the simplified expression is $\frac{16}{x^4}$ .