Select the correct answer.

Which property of exponents must be used first to solve this expression?

[tex] \left(x y^2\right)^{\frac{1}{3}} [/tex]

A. [tex] \frac{a^m}{a^n}=a^{m-n} [/tex]

B. [tex] a^m a^n=a^{m+n} [/tex]

C. [tex] \left(\frac{a}{b}\right)^m=\frac{a^m}{b^m} [/tex]

D. [tex] (a b)^n=a^n b^n [/tex]



Answer :

To simplify the expression [tex]\(\left(x y^2\right)^{\frac{1}{3}}\)[/tex], you need to apply the properties of exponents in the correct order to break it down step-by-step.

In this specific problem, you need to raise both the [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex] inside the parentheses to the power of [tex]\(\frac{1}{3}\)[/tex] because this encompasses the entire product within the parentheses.

Therefore, the property of exponents you should use first is:

D. [tex]\((a b)^n = a^n b^n\)[/tex]

This property allows you to distribute the exponent [tex]\(\frac{1}{3}\)[/tex] to both [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex] individually:
[tex]$ \left(x y^2\right)^{\frac{1}{3}} = x^{\frac{1}{3}} (y^2)^{\frac{1}{3}} $[/tex]