Let's match each expression on the left with its equivalent expression on the right by analyzing the given results.
### Expression Matching Process:
1. Expression: [tex]\(\sqrt[3]{-48 a^6 b^9}\)[/tex]
- Equivalent Expression: [tex]\(-2 a^2 b^2 \sqrt[3]{6 a b^2}\)[/tex]
Reasoning: This equivalency matches with the terms and radicals observed:
- [tex]\(-48 a^6 b^9\)[/tex] simplifies to [tex]\((-2 a^2 b^2) \cdot 6 a b^2\)[/tex], and taking the cube root results in the expression on the right side.
2. Expression: [tex]\(\sqrt{48 a^6 b^9}\)[/tex]
- Equivalent Expression: [tex]\(4 \left|a^3\right| b^4 \sqrt{3 b}\)[/tex]
Reasoning: Examining the mathematical expressions:
- [tex]\(48 a^6 b^9\)[/tex] simplifies with the factorization such that the terms match.
- The square root of [tex]\(a^6\)[/tex] is [tex]\(|a^3|\)[/tex] and [tex]\(\sqrt{48 b^9 / a^6}\)[/tex] leads to [tex]\(4 \sqrt{3 b}\)[/tex].
3. Expression: [tex]\(\sqrt[3]{-48 a^7 b^8}\)[/tex]
- Equivalent Expression: [tex]\(2 a^2 b^3 \sqrt[3]{-6}\)[/tex]
Reasoning: Applying the same logic:
- Breaking down [tex]\(-48 a^7 b^8\)[/tex] results in similar terms that when cube rooted, form [tex]\(2 a^2 b^3 \sqrt[3]{-6}\)[/tex].
Hence, these are the correctly matched pairs based on the given analysis:
- [tex]\(\sqrt[3]{-48 a^6 b^9}\)[/tex] matches with [tex]\(-2 a^2 b^2 \sqrt[3]{6 a b^2}\)[/tex].
- [tex]\(\sqrt{48 a^6 b^9}\)[/tex] matches with [tex]\(4\left|a^3\right| b^4 \sqrt{3 b}\)[/tex].
- [tex]\(\sqrt[3]{-48 a^7 b^8}\)[/tex] matches with [tex]\(2 a^2 b^3 \sqrt[3]{-6}\)[/tex].
### Final Matches
[tex]$
\begin{array}{ll}
\sqrt[3]{-48 a^6 b^9} & -2 a^2 b^2 \sqrt[3]{6 a b^2} \\
\sqrt{48 a^6 b^9} & 4 \left|a^3\right| b^4 \sqrt{3 b} \\
\sqrt[3]{-48 a^7 b^8} & 2 a^2 b^3 \sqrt[3]{-6}
\end{array}
$[/tex]
