Solve (sqrt{125/8}-1/2sqrt{20})divsqrt{5}

$$( \sqrt{ \frac{ 125 }{ 8 } } - \frac{ 1 }{ 2 } \sqrt{ 20 } ) \div \sqrt{ 5 }$$

Evaluate

$$\frac{5\sqrt{2}}{4}-1\approx 0.767766953$$

Short Solution Steps

Rewrite the square root of the division \sqrt{\frac{125}{8}} as the division of square roots \frac{\sqrt{125}}{\sqrt{8}}.

$$\frac{\frac{\sqrt{125}}{\sqrt{8}}-\frac{1}{2}\sqrt{20}}{\sqrt{5}}$$

Factor 125=5^{2}\times 5. Rewrite the square root of the product \sqrt{5^{2}\times 5} as the product of square roots \sqrt{5^{2}}\sqrt{5}. Take the square root of 5^{2}.

$$\frac{\frac{5\sqrt{5}}{\sqrt{8}}-\frac{1}{2}\sqrt{20}}{\sqrt{5}}$$

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}.

$$\frac{\frac{5\sqrt{5}}{2\sqrt{2}}-\frac{1}{2}\sqrt{20}}{\sqrt{5}}$$

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

$$\frac{\frac{5\sqrt{5}\sqrt{2}}{2\left(\sqrt{2}\right)^{2}}-\frac{1}{2}\sqrt{20}}{\sqrt{5}}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{5\sqrt{5}\sqrt{2}}{2\times 2}-\frac{1}{2}\sqrt{20}}{\sqrt{5}}$$

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

$$\frac{\frac{5\sqrt{10}}{2\times 2}-\frac{1}{2}\sqrt{20}}{\sqrt{5}}$$

Multiply 2 and 2 to get 4.

$$\frac{\frac{5\sqrt{10}}{4}-\frac{1}{2}\sqrt{20}}{\sqrt{5}}$$

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}.

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

Cancel out 2 and 2.

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

To add or subtract expressions, expand them to make their denominators the same. Multiply \sqrt{5} times \frac{4}{4}.

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

Since \frac{5\sqrt{10}}{4} and \frac{4\sqrt{5}}{4} have the same denominator, subtract them by subtracting their numerators.

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

Express \frac{\frac{5\sqrt{10}-4\sqrt{5}}{4}}{\sqrt{5}} as a single fraction.

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

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

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

The square of \sqrt{5} is 5.

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

Multiply 4 and 5 to get 20.

$$\frac{\left(5\sqrt{10}-4\sqrt{5}\right)\sqrt{5}}{20}$$

Use the distributive property to multiply 5\sqrt{10}-4\sqrt{5} by \sqrt{5}.

$$\frac{5\sqrt{10}\sqrt{5}-4\left(\sqrt{5}\right)^{2}}{20}$$

Factor 10=5\times 2. Rewrite the square root of the product \sqrt{5\times 2} as the product of square roots \sqrt{5}\sqrt{2}.

$$\frac{5\sqrt{5}\sqrt{2}\sqrt{5}-4\left(\sqrt{5}\right)^{2}}{20}$$

Multiply \sqrt{5} and \sqrt{5} to get 5.

$$\frac{5\times 5\sqrt{2}-4\left(\sqrt{5}\right)^{2}}{20}$$

Multiply 5 and 5 to get 25.

$$\frac{25\sqrt{2}-4\left(\sqrt{5}\right)^{2}}{20}$$

The square of \sqrt{5} is 5.

$$\frac{25\sqrt{2}-4\times 5}{20}$$

Multiply -4 and 5 to get -20.

$$\frac{25\sqrt{2}-20}{20}$$

Factor

$$\frac{5 \sqrt{2} - 4}{4} = 0.7677669529663689$$