To determine which of the given expressions is equivalent to [tex]\(\left(x^3 y \right)^2\)[/tex], we will simplify the expression step-by-step.
First, let's rewrite the given expression:
[tex]\[
\left(x^3 y \right)^2
\][/tex]
When raising a product to a power, we apply the power to each factor inside the parentheses. That means we will raise each term inside the parentheses [tex]\((x^3 \text{ and } y)\)[/tex] to the power of 2 separately. Using the property [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex], we get:
[tex]\[
\left(x^3 y \right)^2 = (x^3)^2 \cdot (y)^2
\][/tex]
Next, we apply the exponent to each individual factor:
[tex]\[
(x^3)^2 \cdot (y)^2
\][/tex]
Now, apply the exponentiation rule [tex]\((a^m)^n = a^{m \cdot n}\)[/tex] to each term:
[tex]\[
(x^3)^2 = x^{3 \cdot 2} = x^6
\][/tex]
[tex]\[
(y)^2 = y^{1 \cdot 2} = y^2
\][/tex]
Combining these results, we get:
[tex]\[
x^6 \cdot y^2 = x^6 y^2
\][/tex]
Therefore, the given expression [tex]\(\left(x^3 y \right)^2\)[/tex] simplifies to [tex]\(x^6 y^2\)[/tex].
So, the correct equivalent expression is:
[tex]\[
\boxed{x^6 y^2}
\][/tex]
From the provided choices:
A) [tex]\(x^8 y^8\)[/tex]
B) [tex]\(x^6 y^3\)[/tex]
C) [tex]\(x^6 y^2\)[/tex]
D) [tex]\(x^5 y^2\)[/tex]
The correct answer is:
[tex]\[
\boxed{C}
\][/tex]
