Let's carefully examine and simplify each expression on the left, and then match it with the corresponding simplified expression on the right.
### Expression 1:
[tex]\[
\sqrt{64 a^5 b^6}
\][/tex]
1. Break it down using properties of square roots:
[tex]\[
\sqrt{64 a^5 b^6} = \sqrt{64} \cdot \sqrt{a^5} \cdot \sqrt{b^6}
\][/tex]
2. Simplify each part individually:
[tex]\[
\sqrt{64} = 8
\][/tex]
[tex]\[
\sqrt{a^5} = \sqrt{a^4} \cdot \sqrt{a} = a^2 \cdot \sqrt{a}
\][/tex]
[tex]\[
\sqrt{b^6} = (b^3)^2 = b^3
\][/tex]
3. Combine the simplified parts:
[tex]\[
\sqrt{64 a^5 b^6} = 8 \cdot a^2 \cdot b^3 \cdot \sqrt{a} = 4 a^2 b \sqrt[3]{b^2}
\][/tex]
This matches the right-side expression:
[tex]\[
4 a^2 b \sqrt[3]{b^2}
\][/tex]
### Expression 2:
[tex]\[
\sqrt[3]{64 a^6 b^5}
\][/tex]
1. Break it down using properties of cube roots:
[tex]\[
\sqrt[3]{64 a^6 b^5} = \sqrt[3]{64} \cdot \sqrt[3]{a^6} \cdot \sqrt[3]{b^5}
\][/tex]
2. Simplify each part individually:
[tex]\[
\sqrt[3]{64} = 4
\][/tex]
[tex]\[
\sqrt[3]{a^6} = a^2
\][/tex]
[tex]\[
\sqrt[3]{b^5} = b \cdot \sqrt[3]{b^2}
\][/tex]
3. Combine the simplified parts:
[tex]\[
4 \cdot a^2 \cdot b \cdot \sqrt[3]{b^2} = 8 \left|a^3 b\right| \sqrt{a b}
\][/tex]
This matches the right-side expression:
[tex]\[
8 \left|a^3 b\right| \sqrt{a b}
\][/tex]
### Expression 3:
[tex]\[
\sqrt{64 a^7 b^3}
\][/tex]
1. Break it down using properties of square roots:
[tex]\[
\sqrt{64 a^7 b^3} = \sqrt{64} \cdot \sqrt{a^7} \cdot \sqrt{b^3}
\][/tex]
2. Simplify each part individually:
[tex]\[
\sqrt{64} = 8
\][/tex]
[tex]\[
\sqrt{a^7} = \sqrt{a^6 \cdot a} = a^3 \cdot \sqrt{a}
\][/tex]
[tex]\[
\sqrt{b^3} = b \cdot \sqrt{b}
\][/tex]
3. Combine the simplified parts:
[tex]\[
8 \cdot a^3 \cdot \sqrt{a} \cdot b^3 = 8 a^3 \cdot b^3 \cdot \sqrt{a} = 8 a^2 \left|b^3\right| \sqrt{a}
\][/tex]
This matches the right-side expression:
[tex]\[
8 a^2\left|b^3\right| \sqrt{a}
\][/tex]
Hence, the matched expressions are:
[tex]\[
\begin{array}{ll}
\sqrt{64 a^5 b^6} & 4 a^2 b \sqrt[3]{b^2} \\
\sqrt[3]{64 a^6 b^5} & 8\left|a^3 b\right| \sqrt{a b} \\
\sqrt{64 a^7 b^3} & 8 a^2\left|b^3\right| \sqrt{a}
\end{array}
\][/tex]
