Solve frac{frac{sqrt{2000}*sqrt{1/2}}{2}}{pi}

$$\frac{ \frac{ \sqrt{ 2000 } \times \sqrt{ \frac{ 1 }{ 2 } } }{ 2 } }{ \pi }$$

Evaluate

$$\frac{5\sqrt{10}}{\pi }\approx 5.03292121$$

Solution Steps

Express \frac{\frac{\sqrt{2000}\sqrt{\frac{1}{2}}}{2}}{\pi } as a single fraction.

$$\frac{\sqrt{2000}\sqrt{\frac{1}{2}}}{2\pi }$$

Factor 2000=20^{2}\times 5. Rewrite the square root of the product \sqrt{20^{2}\times 5} as the product of square roots \sqrt{20^{2}}\sqrt{5}. Take the square root of 20^{2}.

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

Rewrite the square root of the division \sqrt{\frac{1}{2}} as the division of square roots \frac{\sqrt{1}}{\sqrt{2}}.

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

Calculate the square root of 1 and get 1.

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

Rationalize the denominator of \frac{1}{\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

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

The square of \sqrt{2} is 2.

$$\frac{20\sqrt{5}\times \frac{\sqrt{2}}{2}}{2\pi }$$

Cancel out 2, the greatest common factor in 20 and 2.

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

Cancel out 2 in both numerator and denominator.

$$\frac{5\sqrt{2}\sqrt{5}}{\pi }$$

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

$$\frac{5\sqrt{10}}{\pi }$$