Question
Prove that:
$\Big(\sqrt{3\times5^{-3}}\div\sqrt[3]{3^{-1}}\sqrt{5}\times\sqrt[6]{3\times5^6}=\frac{3}{5}\Big)$
$\Big(\sqrt{3\times5^{-3}}\div\sqrt[3]{3^{-1}}\sqrt{5}\times\sqrt[6]{3\times5^6}=\frac{3}{5}\Big)$
✓
Answer
We have to prove that $\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=\frac{3}{5}$
By using rational exponent $\text{a}^{-\text{n}}=\frac{1}{\text{a}^{\text{n}}}$ we get,
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=\frac{\sqrt{3\times\frac{1}{5^3}}}{\sqrt[3]{\frac{1}{3}}\sqrt{5}}\times\sqrt[6]{3\times5^6}$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=\frac{3^\frac{1}{2}\times\frac{1}{5^{3\times\frac{1}{2}}}}{\frac{1}{3^{\frac{1}{3}}}\times5^\frac{1}{2}}\times3^\frac{1}{6}\times5^{6\times\frac{1}{6}}$
$=\frac{\frac{3^\frac{1}{2}}{5^{\frac{3}{2}}}}{\frac{5^{\frac{1}{5}}}{3^\frac{1}{3}}}\times3^\frac{1}{6}\times5^1$
$=\frac{3^\frac{1}{2}}{5^\frac{3}{2}}\times\frac{3^\frac{1}{3}}{5^\frac{1}{2}}\times3^\frac{1}{6}\times5^1$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=3^\frac{1}{3}\times3^\frac{1}{3}\times5^{-\frac{3}{2}}\times5^{-\frac{1}{2}}\times3^{\frac{1}{6}}\times5^1$
$=3^{\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}\times5^{-\frac{3}{2}-\frac{1}{2}+1}$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=3^{\frac{1\times3}{2\times3}+\frac{1\times2}{3\times2}+\frac{1}{6}}\times5^{-\frac{3}{2}-\frac{1}{2}+\frac{1}{6}}$
$=3^{\frac{3+2+1}{6}}\times5^{\frac{-3-1+2}{2}}$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}3^\frac{6}{6}\times5^\frac{2}{2}=3^1\times5^{-1}=\frac{3}{5}$
Hence, $\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=\frac{3}{5}$
By using rational exponent $\text{a}^{-\text{n}}=\frac{1}{\text{a}^{\text{n}}}$ we get,
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=\frac{\sqrt{3\times\frac{1}{5^3}}}{\sqrt[3]{\frac{1}{3}}\sqrt{5}}\times\sqrt[6]{3\times5^6}$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=\frac{3^\frac{1}{2}\times\frac{1}{5^{3\times\frac{1}{2}}}}{\frac{1}{3^{\frac{1}{3}}}\times5^\frac{1}{2}}\times3^\frac{1}{6}\times5^{6\times\frac{1}{6}}$
$=\frac{\frac{3^\frac{1}{2}}{5^{\frac{3}{2}}}}{\frac{5^{\frac{1}{5}}}{3^\frac{1}{3}}}\times3^\frac{1}{6}\times5^1$
$=\frac{3^\frac{1}{2}}{5^\frac{3}{2}}\times\frac{3^\frac{1}{3}}{5^\frac{1}{2}}\times3^\frac{1}{6}\times5^1$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=3^\frac{1}{3}\times3^\frac{1}{3}\times5^{-\frac{3}{2}}\times5^{-\frac{1}{2}}\times3^{\frac{1}{6}}\times5^1$
$=3^{\frac{1}{2}+\frac{1}{3}+\frac{1}{6}}\times5^{-\frac{3}{2}-\frac{1}{2}+1}$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=3^{\frac{1\times3}{2\times3}+\frac{1\times2}{3\times2}+\frac{1}{6}}\times5^{-\frac{3}{2}-\frac{1}{2}+\frac{1}{6}}$
$=3^{\frac{3+2+1}{6}}\times5^{\frac{-3-1+2}{2}}$
$\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}3^\frac{6}{6}\times5^\frac{2}{2}=3^1\times5^{-1}=\frac{3}{5}$
Hence, $\frac{\sqrt{3\times5^{-3}}}{\sqrt[3]{3^{-1}\sqrt{5}}}\times\sqrt[6]{3\times5^6}=\frac{3}{5}$
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