which expression is equivalent to $\frac{sqrt{10}}{sqrt4{8}}$?\n$\frac{sqrt4{200}}{2}$\n$\frac{sqrt4{20}}{2}$…

Explanation:

Step1: Rationalize the denominator

We have $\frac{\sqrt{10}}{\sqrt[4]{8}}$. First, rewrite the radicals in exponent - form. $\sqrt{10}=10^{\frac{1}{2}}$ and $\sqrt[4]{8}=8^{\frac{1}{4}}=(2^{3})^{\frac{1}{4}} = 2^{\frac{3}{4}}$. Multiply the numerator and denominator by $\sqrt[4]{8^{3}}$ to rationalize the denominator. $\frac{\sqrt{10}}{\sqrt[4]{8}}\times\frac{\sqrt[4]{8^{3}}}{\sqrt[4]{8^{3}}}=\frac{\sqrt{10}\times\sqrt[4]{8^{3}}}{\sqrt[4]{8^{4}}}$

Step2: Simplify the denominator

Since $\sqrt[4]{8^{4}} = 8$, the expression becomes $\frac{\sqrt{10}\times\sqrt[4]{512}}{8}$. Now, $\sqrt{10}=10^{\frac{1}{2}}$ and $\sqrt[4]{512}=512^{\frac{1}{4}}=(2^{9})^{\frac{1}{4}}=2^{\frac{9}{4}}$. Also, $\sqrt{10}\times\sqrt[4]{512}=10^{\frac{1}{2}}\times2^{\frac{9}{4}}=(2\times5)^{\frac{1}{2}}\times2^{\frac{9}{4}}=2^{\frac{1}{2}}\times5^{\frac{1}{2}}\times2^{\frac{9}{4}}=2^{\frac{1}{2}+\frac{9}{4}}\times5^{\frac{1}{2}}=2^{\frac{2 + 9}{4}}\times5^{\frac{1}{2}}=2^{\frac{11}{4}}\times5^{\frac{1}{2}}$. Another way: $\frac{\sqrt{10}}{\sqrt[4]{8}}=\frac{\sqrt{10}}{\sqrt[4]{8}}\times\frac{\sqrt[4]{2}}{\sqrt[4]{2}}=\frac{\sqrt{10}\times\sqrt[4]{2}}{\sqrt[4]{16}}=\frac{\sqrt{10}\times\sqrt[4]{2}}{2}$. $\sqrt{10}\times\sqrt[4]{2}=(10)^{\frac{1}{2}}\times(2)^{\frac{1}{4}}=(2\times5)^{\frac{1}{2}}\times(2)^{\frac{1}{4}}=2^{\frac{1}{2}}\times5^{\frac{1}{2}}\times2^{\frac{1}{4}}=2^{\frac{1}{2}+\frac{1}{4}}\times5^{\frac{1}{2}}=2^{\frac{2 + 1}{4}}\times5^{\frac{1}{2}}=2^{\frac{3}{4}}\times5^{\frac{1}{2}}=\sqrt[4]{8}\times\sqrt{5}$. $\frac{\sqrt{10}}{\sqrt[4]{8}}=\frac{\sqrt{10}}{\sqrt[4]{8}}\times\frac{\sqrt[4]{8^{3}}}{\sqrt[4]{8^{3}}}=\frac{\sqrt{10}\times\sqrt[4]{512}}{8}$. $\sqrt{10}\times\sqrt[4]{512}=\sqrt{10}\times\sqrt[4]{2^{9}}=\sqrt{10}\times2^{\frac{9}{4}}$. Let's simplify $\frac{\sqrt{10}}{\sqrt[4]{8}}$ in a different way: $\frac{\sqrt{10}}{\sqrt[4]{8}}=\frac{\sqrt{10}}{\sqrt[4]{8}}\times\frac{\sqrt[4]{2}}{\sqrt[4]{2}}=\frac{\sqrt{10\times2}}{\sqrt[4]{16}}=\frac{\sqrt{20}}{2}=\frac{\sqrt{4\times5}}{2}=\frac{2\sqrt{5}}{2}=\sqrt{5}$ Now, $\frac{\sqrt[4]{20}}{2}=\frac{\sqrt[4]{4\times5}}{2}=\frac{\sqrt{2}\times\sqrt[4]{5}}{2}$ $\frac{\sqrt{10}}{\sqrt[4]{8}}=\frac{\sqrt{10}}{\sqrt[4]{8}}\times\frac{\sqrt[4]{2}}{\sqrt[4]{2}}=\frac{\sqrt{10\times2}}{\sqrt[4]{16}}=\frac{\sqrt{20}}{2}$

Answer:

$\frac{\sqrt[4]{20}}{2}$