Sure, let's analyze this expression step-by-step.
Given expression:
[tex]\[
\left( x y^2 \right)^{\frac{1}{3}}
\][/tex]
We need to determine which property of exponents should be applied first to simplify this expression. Let's examine each property listed:
A. [tex]\((a b)^n = a^n b^n\)[/tex]
This property states that when a product inside parentheses is raised to an exponent, we can distribute the exponent to each factor inside the parentheses.
B. [tex]\(a^m a^n = a^{m+n}\)[/tex]
This property is used when multiplying like bases with different exponents. It combines the exponents by adding them.
C. [tex]\(\left(\frac{a}{b}\right)^m = \frac{a^m}{b^m}\)[/tex]
This property is applied when dividing bases inside parentheses raised to an exponent. We can distribute the exponent to both the numerator and the denominator.
D. [tex]\(\frac{a^m}{a^m} = a^{m-n}\)[/tex]
This property is used when dividing like bases with different exponents. It combines the exponents by subtracting them.
Given our expression, [tex]\(\left( x y^2 \right)^{\frac{1}{3}}\)[/tex], we have a product inside the parentheses that is raised to an exponent.
Applying property (A), we distribute the exponent to each factor in the product:
[tex]\[
(x y^2)^{\frac{1}{3}} = x^{\frac{1}{3}} (y^2)^{\frac{1}{3}}
\][/tex]
So, the first property we use is:
A. [tex]\((a b)^n = a^n b^n\)[/tex]
Thus, the correct answer is:
A. [tex]\((a b)^n = a^n b^n\)[/tex]
