To simplify the expression [tex]\(\sqrt[3]{125 x^2 y^7}\)[/tex] using rational exponent properties and the definition of a radical in terms of exponents, we can break it down step-by-step as follows:
[tex]\[
\begin{array}{|l|l|}
\hline \text{Step 1} & \sqrt[3]{125 x^2 y^7} \\
\hline \text{Step 2} & 125^{1/3} (x^2)^{1/3} (y^7)^{1/3} \\
\hline \text{Step 3} & 5 \cdot x^{2/3} \cdot y^{7/3} \\
\hline \text{Step 4} & 5 \cdot x^{2/3} \cdot y^{2 + 1/3} \\
\hline \text{Step 5} & 5 \cdot x^{2/3} \cdot y^2 \cdot y^{1/3} \\
\hline \text{Step 6} & 5y^2 \cdot x^{2/3} \cdot y^{1/3} \\
\hline \text{Step 7} & 5y^2 \cdot xy^{1/3} \\
\hline
\end{array}
\][/tex]
Explanation of each step:
- Step 1: Start with the given expression [tex]\(\sqrt[3]{125 x^2 y^7}\)[/tex].
- Step 2: Express the cube root in terms of rational exponents: [tex]\(125^{1/3} (x^2)^{1/3} (y^7)^{1/3}\)[/tex].
- Step 3: Simplify each term separately. [tex]\(125^{1/3} = 5\)[/tex], [tex]\((x^2)^{1/3} = x^{2/3}\)[/tex], [tex]\((y^7)^{1/3} = y^{7/3}\)[/tex].
- Step 4: Decompose [tex]\(y^{7/3}\)[/tex] into [tex]\(y^2 \cdot y^{1/3}\)[/tex] by writing [tex]\(7/3\)[/tex] as [tex]\(2 + 1/3\)[/tex].
- Step 5: Combine the terms to separate out [tex]\(y^2\)[/tex] and [tex]\(y^{1/3}\)[/tex].
- Step 6: Rewrite the combined term for clarity: [tex]\(5y^2 \cdot x^{2/3} \cdot y^{1/3}\)[/tex].
- Step 7: Re-arrange the terms, if needed, for standard form: [tex]\(5y^2 \cdot x^{2/3} \cdot y^{1/3}\)[/tex].
