Let's determine the expression equivalent to [tex]\(\left(x^{27} y\right)^\frac{1}{3}\)[/tex] step-by-step.
1. Understand the given expression: [tex]\(\left(x^{27} y\right)^\frac{1}{3}\)[/tex].
2. Apply the property of exponents: The property [tex]\((a \cdot b)^n = a^n \cdot b^n\)[/tex] allows us to separate the base terms inside the parentheses.
[tex]\[
\left(x^{27} y\right)^\frac{1}{3} = \left(x^{27}\right)^\frac{1}{3} \cdot \left(y\right)^\frac{1}{3}
\][/tex]
3. Simplify the first part: [tex]\(\left(x^{27}\right)^\frac{1}{3}\)[/tex]
- Use the exponent rule [tex]\( (a^m)^n = a^{m \cdot n} \)[/tex].
[tex]\[
\left(x^{27}\right)^\frac{1}{3} = x^{27 \cdot \frac{1}{3}} = x^9
\][/tex]
4. Simplify the second part: [tex]\(\left(y\right)^\frac{1}{3}\)[/tex]
- This can be written as the cube root of [tex]\( y \)[/tex], denoted as [tex]\( \sqrt[3]{y} \)[/tex].
[tex]\[
\left(y\right)^\frac{1}{3} = \sqrt[3]{y}
\][/tex]
5. Combine the results: Combine the simplified parts to get the final expression.
[tex]\[
\left(x^{27} y\right)^\frac{1}{3} = x^9 \cdot \sqrt[3]{y}
\][/tex]
So, the equivalent expression is:
[tex]\[
x^9 (\sqrt[3]{y})
\][/tex]
Therefore, the correct choice is [tex]\( \boxed{x^9 (\sqrt[3]{y})} \)[/tex].
