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

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

A. [tex]\(a^m a^n = a^{m+n}\)[/tex]

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

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

D. [tex]\(\frac{a^m}{a^n} = a^{m-n}\)[/tex]



Answer :

To determine which property of exponents to use first in the expression [tex]\(\left(x y^2\right)^{\frac{1}{3}}\)[/tex], let's analyze the expression step by step.

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

We need to distribute the exponent [tex]\(\frac{1}{3}\)[/tex] to both [tex]\(x\)[/tex] and [tex]\(y^2\)[/tex] inside the parentheses. This is done using the property of exponents for a product. Specifically, we use:

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

Applying this property to [tex]\(\left(x y^2\right)^{\frac{1}{3}}\)[/tex], we get:

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

Therefore, the correct property of exponents that must be used first is:

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

Hence, the correct answer is option B.