To determine which expression is equivalent to [tex]\(\left(x^{27} y\right)^{\frac{1}{3}}\)[/tex], we need to simplify the given expression step-by-step and then compare the result with the given options.
1. Original Expression:
[tex]\[\left(x^{27} y\right)^{\frac{1}{3}}\][/tex]
2. Apply the exponent rule:
The exponent rule states that [tex]\((a \cdot b)^c = a^c \cdot b^c\)[/tex].
Hence,
[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 terms inside:
For [tex]\( \left(x^{27}\right)^{\frac{1}{3}} \)[/tex]:
[tex]\[(x^{27})^{\frac{1}{3}} = x^{27 \cdot \frac{1}{3}} = x^9\][/tex]
For [tex]\( \left(y\right)^{\frac{1}{3}} \)[/tex]:
[tex]\[\left(y\right)^{\frac{1}{3}} = y^{\frac{1}{3}}\][/tex]
4. Combine the simplified terms:
So the simplified form of the original expression is:
[tex]\[x^9 \cdot y^{\frac{1}{3}}\][/tex]
5. Compare with the given options:
[tex]\[
\begin{aligned}
&\text{Option 1: } x^3 (\sqrt[3]{y}) = x^3 y^{\frac{1}{3}} \\
&\text{Option 2: } x^9 (\sqrt[3]{y}) = x^9 y^{\frac{1}{3}} \\
&\text{Option 3: } x^{27} (\sqrt[3]{y}) = x^{27} y^{\frac{1}{3}} \\
&\text{Option 4: } x^{24} (\sqrt[3]{y}) = x^{24} y^{\frac{1}{3}} \\
\end{aligned}
\][/tex]
Clearly, by comparison, the equivalent expression to [tex]\(\left(x^{27} y\right)^{\frac{1}{3}}\)[/tex] is:
[tex]\[
x^9 (\sqrt[3]{y})
\][/tex]
Therefore, the answer to the question is the second option:
[tex]\[ x^9(\sqrt[3]{y}) \][/tex]
