To determine which expression is equivalent to [tex]\(\sqrt[3]{27 x^5 y^8}\)[/tex], let's break down and simplify the given expression step-by-step.
Given expression: [tex]\(\sqrt[3]{27 x^5 y^8}\)[/tex]
Step 1: Break the expression under the cube root into separate parts that we can apply the cube root to individually:
[tex]\[
\sqrt[3]{27 x^5 y^8} = \sqrt[3]{27} \cdot \sqrt[3]{x^5} \cdot \sqrt[3]{y^8}
\][/tex]
Step 2: Simplify each part individually:
1. [tex]\(\sqrt[3]{27} = 3\)[/tex]
2. [tex]\(\sqrt[3]{x^5} = x^{5/3}\)[/tex]
3. [tex]\(\sqrt[3]{y^8} = y^{8/3}\)[/tex]
Thus, the expression becomes:
[tex]\[
3 \cdot x^{5/3} \cdot y^{8/3}
\][/tex]
Step 3: Combine all the simplified parts into one expression:
[tex]\[
3 x^{5/3} y^{8/3}
\][/tex]
Now, compare this resulting expression with the given options:
1. [tex]\(3 x^{\frac{5}{5}} y^{\frac{8}{3}}\)[/tex]
2. [tex]\(9 x^{\frac{5}{3}} y^{\frac{8}{3}}\)[/tex]
3. [tex]\(3 x^{\frac{3}{3}} y^{\frac{3}{8}}\)[/tex]
4. [tex]\(9 x^3 y^{\frac{3}{8}}\)[/tex]
The correct expression [tex]\(3 x^{5/3} y^{8/3}\)[/tex] matches the first option:
Answer: [tex]\(3 x^{\frac{5}{3}} y^{\frac{8}{3}}\)[/tex]
