Solve sqrt[8]{5/4}+3sqrt{20}-10sqrt{1/5}

$$\sqrt[ 8 ]{ \frac{ 5 }{ 4 } } +3 \sqrt{ 20 } -10 \sqrt{ \frac{ 1 }{ 5 } }$$

Evaluate

$$\frac{4^{\frac{7}{8}}\sqrt[8]{5}}{4}+4\sqrt{5}\approx 9.972557504$$

Solution Steps

Factor 20=2^{2}\times 5. Rewrite the square root of the product \sqrt{2^{2}\times 5} as the product of square roots \sqrt{2^{2}}\sqrt{5}. Take the square root of 2^{2}.

$$\sqrt[8]{\frac{5}{4}}+3\times 2\sqrt{5}-10\sqrt{\frac{1}{5}}$$

Multiply 3 and 2 to get 6.

$$\sqrt[8]{\frac{5}{4}}+6\sqrt{5}-10\sqrt{\frac{1}{5}}$$

Rewrite the square root of the division \sqrt{\frac{1}{5}} as the division of square roots \frac{\sqrt{1}}{\sqrt{5}}.

$$\sqrt[8]{\frac{5}{4}}+6\sqrt{5}-10\times \frac{\sqrt{1}}{\sqrt{5}}$$

Calculate the square root of 1 and get 1.

$$\sqrt[8]{\frac{5}{4}}+6\sqrt{5}-10\times \frac{1}{\sqrt{5}}$$

Rationalize the denominator of \frac{1}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\sqrt[8]{\frac{5}{4}}+6\sqrt{5}-10\times \frac{\sqrt{5}}{\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\sqrt[8]{\frac{5}{4}}+6\sqrt{5}-10\times \frac{\sqrt{5}}{5}$$

Cancel out 5, the greatest common factor in 10 and 5.

$$\sqrt[8]{\frac{5}{4}}+6\sqrt{5}-2\sqrt{5}$$

Combine 6\sqrt{5} and -2\sqrt{5} to get 4\sqrt{5}.

$$\sqrt[8]{\frac{5}{4}}+4\sqrt{5}$$