Solve sqrt{5/8}*sqrt{1-1/2+1/10-1/5}?

$$\sqrt { \frac { 5 } { 8 } } \cdot \sqrt { 1 - \frac { 1 } { 2 } + \frac { 1 } { 10 } - \frac { 1 } { 5 } } ?$$

Evaluate

$$\frac{1}{2}=0.5$$

Solution Steps

Rewrite the square root of the division \sqrt{\frac{5}{8}} as the division of square roots \frac{\sqrt{5}}{\sqrt{8}}.

$$\frac{\sqrt{5}}{\sqrt{8}}\sqrt{1-\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

Factor 8=2^{2}\times 2. Rewrite the square root of the product \sqrt{2^{2}\times 2} as the product of square roots \sqrt{2^{2}}\sqrt{2}. Take the square root of 2^{2}.

$$\frac{\sqrt{5}}{2\sqrt{2}}\sqrt{1-\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

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

$$\frac{\sqrt{5}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}\sqrt{1-\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{5}\sqrt{2}}{2\times 2}\sqrt{1-\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

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

$$\frac{\sqrt{10}}{2\times 2}\sqrt{1-\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

Multiply 2 and 2 to get 4.

$$\frac{\sqrt{10}}{4}\sqrt{1-\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

Convert 1 to fraction \frac{2}{2}.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{2}{2}-\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

Since \frac{2}{2} and \frac{1}{2} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{2-1}{2}+\frac{1}{10}-\frac{1}{5}}$$

Subtract 1 from 2 to get 1.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{1}{2}+\frac{1}{10}-\frac{1}{5}}$$

Least common multiple of 2 and 10 is 10. Convert \frac{1}{2} and \frac{1}{10} to fractions with denominator 10.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{5}{10}+\frac{1}{10}-\frac{1}{5}}$$

Since \frac{5}{10} and \frac{1}{10} have the same denominator, add them by adding their numerators.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{5+1}{10}-\frac{1}{5}}$$

Add 5 and 1 to get 6.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{6}{10}-\frac{1}{5}}$$

Reduce the fraction \frac{6}{10} to lowest terms by extracting and canceling out 2.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{3}{5}-\frac{1}{5}}$$

Since \frac{3}{5} and \frac{1}{5} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{3-1}{5}}$$

Subtract 1 from 3 to get 2.

$$\frac{\sqrt{10}}{4}\sqrt{\frac{2}{5}}$$

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

$$\frac{\sqrt{10}}{4}\times \frac{\sqrt{2}}{\sqrt{5}}$$

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

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

The square of \sqrt{5} is 5.

$$\frac{\sqrt{10}}{4}\times \frac{\sqrt{2}\sqrt{5}}{5}$$

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

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

Multiply \frac{\sqrt{10}}{4} times \frac{\sqrt{10}}{5} by multiplying numerator times numerator and denominator times denominator.

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

Multiply \sqrt{10} and \sqrt{10} to get 10.

$$\frac{10}{4\times 5}$$

Multiply 4 and 5 to get 20.

$$\frac{10}{20}$$

Reduce the fraction \frac{10}{20} to lowest terms by extracting and canceling out 10.

$$\frac{1}{2}$$

Factor

$$\frac{1}{2} = 0.5$$