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Sagot :
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]
### 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]
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