Solve frac{10*sqrt{3}}{2*pi}*sqrt{1-1/2*{left(frac{25{7*sqrt{3}}right)}^2}}

$$\frac{ 10 \times \sqrt{ 3 } }{ 2 \times \pi } \times \sqrt{ 1- \frac{ 1 }{ 2 \times { \left( \frac{ 25 }{ 7 \times \sqrt{ 3 } } \right) }^{ 2 } } }$$

Evaluate

$$\frac{\sqrt{6618}}{10\pi }\approx 2.58948565$$

Solution Steps

Cancel out 2 in both numerator and denominator.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{1}{2\times \left(\frac{25}{7\sqrt{3}}\right)^{2}}}$$

Rationalize the denominator of \frac{25}{7\sqrt{3}} by multiplying numerator and denominator by \sqrt{3}.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{1}{2\times \left(\frac{25\sqrt{3}}{7\left(\sqrt{3}\right)^{2}}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{1}{2\times \left(\frac{25\sqrt{3}}{7\times 3}\right)^{2}}}$$

Multiply 7 and 3 to get 21.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{1}{2\times \left(\frac{25\sqrt{3}}{21}\right)^{2}}}$$

To raise \frac{25\sqrt{3}}{21} to a power, raise both numerator and denominator to the power and then divide.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{1}{2\times \frac{\left(25\sqrt{3}\right)^{2}}{21^{2}}}}$$

Express 2\times \frac{\left(25\sqrt{3}\right)^{2}}{21^{2}} as a single fraction.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{1}{\frac{2\times \left(25\sqrt{3}\right)^{2}}{21^{2}}}}$$

Divide 1 by \frac{2\times \left(25\sqrt{3}\right)^{2}}{21^{2}} by multiplying 1 by the reciprocal of \frac{2\times \left(25\sqrt{3}\right)^{2}}{21^{2}}.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{21^{2}}{2\times \left(25\sqrt{3}\right)^{2}}}$$

Calculate 21 to the power of 2 and get 441.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{441}{2\times \left(25\sqrt{3}\right)^{2}}}$$

Expand \left(25\sqrt{3}\right)^{2}.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{441}{2\times 25^{2}\left(\sqrt{3}\right)^{2}}}$$

Calculate 25 to the power of 2 and get 625.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{441}{2\times 625\left(\sqrt{3}\right)^{2}}}$$

The square of \sqrt{3} is 3.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{441}{2\times 625\times 3}}$$

Multiply 625 and 3 to get 1875.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{441}{2\times 1875}}$$

Multiply 2 and 1875 to get 3750.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{441}{3750}}$$

Reduce the fraction \frac{441}{3750} to lowest terms by extracting and canceling out 3.

$$\frac{5\sqrt{3}}{\pi }\sqrt{1-\frac{147}{1250}}$$

Subtract \frac{147}{1250} from 1 to get \frac{1103}{1250}.

$$\frac{5\sqrt{3}}{\pi }\sqrt{\frac{1103}{1250}}$$

Rewrite the square root of the division \sqrt{\frac{1103}{1250}} as the division of square roots \frac{\sqrt{1103}}{\sqrt{1250}}.

$$\frac{5\sqrt{3}}{\pi }\times \frac{\sqrt{1103}}{\sqrt{1250}}$$

Factor 1250=25^{2}\times 2. Rewrite the square root of the product \sqrt{25^{2}\times 2} as the product of square roots \sqrt{25^{2}}\sqrt{2}. Take the square root of 25^{2}.

$$\frac{5\sqrt{3}}{\pi }\times \frac{\sqrt{1103}}{25\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{1103}}{25\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{5\sqrt{3}}{\pi }\times \frac{\sqrt{1103}\sqrt{2}}{25\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{5\sqrt{3}}{\pi }\times \frac{\sqrt{1103}\sqrt{2}}{25\times 2}$$

To multiply \sqrt{1103} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{5\sqrt{3}}{\pi }\times \frac{\sqrt{2206}}{25\times 2}$$

Multiply 25 and 2 to get 50.

$$\frac{5\sqrt{3}}{\pi }\times \frac{\sqrt{2206}}{50}$$

Multiply \frac{5\sqrt{3}}{\pi } times \frac{\sqrt{2206}}{50} by multiplying numerator times numerator and denominator times denominator.

$$\frac{5\sqrt{3}\sqrt{2206}}{\pi \times 50}$$

Cancel out 5 in both numerator and denominator.

$$\frac{\sqrt{3}\sqrt{2206}}{10\pi }$$

To multiply \sqrt{3} and \sqrt{2206}, multiply the numbers under the square root.

$$\frac{\sqrt{6618}}{10\pi }$$