Solve sqrt{frac{4/4.1253}}{sqrt{2/5}}

$$\frac{ \sqrt{ \frac{ 4 }{ 4.125 } 3 } }{ \sqrt{ \frac{ 2 }{ 5 } } }$$

Evaluate

$$\frac{4\sqrt{55}}{11}\approx 2.69679945$$

Solution Steps

Expand \frac{4}{4.125} by multiplying both numerator and the denominator by 1000.

$$\frac{\sqrt{\frac{4000}{4125}\times 3}}{\sqrt{\frac{2}{5}}}$$

Reduce the fraction \frac{4000}{4125} to lowest terms by extracting and canceling out 125.

$$\frac{\sqrt{\frac{32}{33}\times 3}}{\sqrt{\frac{2}{5}}}$$

Express \frac{32}{33}\times 3 as a single fraction.

$$\frac{\sqrt{\frac{32\times 3}{33}}}{\sqrt{\frac{2}{5}}}$$

Multiply 32 and 3 to get 96.

$$\frac{\sqrt{\frac{96}{33}}}{\sqrt{\frac{2}{5}}}$$

Reduce the fraction \frac{96}{33} to lowest terms by extracting and canceling out 3.

$$\frac{\sqrt{\frac{32}{11}}}{\sqrt{\frac{2}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{32}{11}} as the division of square roots \frac{\sqrt{32}}{\sqrt{11}}.

$$\frac{\frac{\sqrt{32}}{\sqrt{11}}}{\sqrt{\frac{2}{5}}}$$

Factor 32=4^{2}\times 2. Rewrite the square root of the product \sqrt{4^{2}\times 2} as the product of square roots \sqrt{4^{2}}\sqrt{2}. Take the square root of 4^{2}.

$$\frac{\frac{4\sqrt{2}}{\sqrt{11}}}{\sqrt{\frac{2}{5}}}$$

Rationalize the denominator of \frac{4\sqrt{2}}{\sqrt{11}} by multiplying numerator and denominator by \sqrt{11}.

$$\frac{\frac{4\sqrt{2}\sqrt{11}}{\left(\sqrt{11}\right)^{2}}}{\sqrt{\frac{2}{5}}}$$

The square of \sqrt{11} is 11.

$$\frac{\frac{4\sqrt{2}\sqrt{11}}{11}}{\sqrt{\frac{2}{5}}}$$

To multiply \sqrt{2} and \sqrt{11}, multiply the numbers under the square root.

$$\frac{\frac{4\sqrt{22}}{11}}{\sqrt{\frac{2}{5}}}$$

Rewrite the square root of the division \sqrt{\frac{2}{5}} as the division of square roots \frac{\sqrt{2}}{\sqrt{5}}.

$$\frac{\frac{4\sqrt{22}}{11}}{\frac{\sqrt{2}}{\sqrt{5}}}$$

Rationalize the denominator of \frac{\sqrt{2}}{\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{\frac{4\sqrt{22}}{11}}{\frac{\sqrt{2}\sqrt{5}}{\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{\frac{4\sqrt{22}}{11}}{\frac{\sqrt{2}\sqrt{5}}{5}}$$

To multiply \sqrt{2} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{\frac{4\sqrt{22}}{11}}{\frac{\sqrt{10}}{5}}$$

Divide \frac{4\sqrt{22}}{11} by \frac{\sqrt{10}}{5} by multiplying \frac{4\sqrt{22}}{11} by the reciprocal of \frac{\sqrt{10}}{5}.

$$\frac{4\sqrt{22}\times 5}{11\sqrt{10}}$$

Rationalize the denominator of \frac{4\sqrt{22}\times 5}{11\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{4\sqrt{22}\times 5\sqrt{10}}{11\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{4\sqrt{22}\times 5\sqrt{10}}{11\times 10}$$

Multiply 4 and 5 to get 20.

$$\frac{20\sqrt{22}\sqrt{10}}{11\times 10}$$

To multiply \sqrt{22} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{20\sqrt{220}}{11\times 10}$$

Multiply 11 and 10 to get 110.

$$\frac{20\sqrt{220}}{110}$$

Factor 220=2^{2}\times 55. Rewrite the square root of the product \sqrt{2^{2}\times 55} as the product of square roots \sqrt{2^{2}}\sqrt{55}. Take the square root of 2^{2}.

$$\frac{20\times 2\sqrt{55}}{110}$$

Multiply 20 and 2 to get 40.

$$\frac{40\sqrt{55}}{110}$$

Divide 40\sqrt{55} by 110 to get \frac{4}{11}\sqrt{55}.

$$\frac{4}{11}\sqrt{55}$$