Solve 1/2*pi*sqrt{2*10^-3*90*10^-9}*sqrt{frac{2*10^-3/90*10^-9-{20}^2}{frac{2*{10}^-3}{90*{10}^-9}}}

$$\frac{ 1 }{ 2 \times \pi \times \sqrt{ 2 \times { 10 }^{ -3 } \times 90 \times { 10 }^{ -9 } } } \times \sqrt{ \frac{ \frac{ 2 \times { 10 }^{ -3 } }{ 90 \times { 10 }^{ -9 } } - { 20 }^{ 2 } }{ \frac{ 2 \times { 10 }^{ -3 } }{ 90 \times { 10 }^{ -9 } } } }$$

Evaluate

$$\frac{5000\sqrt{491}}{3\pi }\approx 11755.4598625$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -3 and -9 to get -12.

$$\frac{1}{2\pi \sqrt{2\times 10^{-12}\times 90}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Calculate 10 to the power of -12 and get \frac{1}{1000000000000}.

$$\frac{1}{2\pi \sqrt{2\times \frac{1}{1000000000000}\times 90}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Multiply 2 and \frac{1}{1000000000000} to get \frac{1}{500000000000}.

$$\frac{1}{2\pi \sqrt{\frac{1}{500000000000}\times 90}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Multiply \frac{1}{500000000000} and 90 to get \frac{9}{50000000000}.

$$\frac{1}{2\pi \sqrt{\frac{9}{50000000000}}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Rewrite the square root of the division \sqrt{\frac{9}{50000000000}} as the division of square roots \frac{\sqrt{9}}{\sqrt{50000000000}}.

$$\frac{1}{2\pi \times \frac{\sqrt{9}}{\sqrt{50000000000}}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Calculate the square root of 9 and get 3.

$$\frac{1}{2\pi \times \frac{3}{\sqrt{50000000000}}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Factor 50000000000=100000^{2}\times 5. Rewrite the square root of the product \sqrt{100000^{2}\times 5} as the product of square roots \sqrt{100000^{2}}\sqrt{5}. Take the square root of 100000^{2}.

$$\frac{1}{2\pi \times \frac{3}{100000\sqrt{5}}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Rationalize the denominator of \frac{3}{100000\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{1}{2\pi \times \frac{3\sqrt{5}}{100000\left(\sqrt{5}\right)^{2}}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

The square of \sqrt{5} is 5.

$$\frac{1}{2\pi \times \frac{3\sqrt{5}}{100000\times 5}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Multiply 100000 and 5 to get 500000.

$$\frac{1}{2\pi \times \frac{3\sqrt{5}}{500000}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Cancel out 500000, the greatest common factor in 2 and 500000.

$$\frac{1}{\frac{3\sqrt{5}}{250000}\pi }\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Express \frac{3\sqrt{5}}{250000}\pi as a single fraction.

$$\frac{1}{\frac{3\sqrt{5}\pi }{250000}}\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Divide 1 by \frac{3\sqrt{5}\pi }{250000} by multiplying 1 by the reciprocal of \frac{3\sqrt{5}\pi }{250000}.

$$\frac{250000}{3\sqrt{5}\pi }\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Rationalize the denominator of \frac{250000}{3\sqrt{5}\pi } by multiplying numerator and denominator by \sqrt{5}.

$$\frac{250000\sqrt{5}}{3\left(\sqrt{5}\right)^{2}\pi }\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

The square of \sqrt{5} is 5.

$$\frac{250000\sqrt{5}}{3\times 5\pi }\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Cancel out 5 in both numerator and denominator.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{2\times 10^{-3}}{90\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{10^{-3}}{45\times 10^{-9}}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{10^{6}}{45}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Calculate 10 to the power of 6 and get 1000000.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{1000000}{45}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Reduce the fraction \frac{1000000}{45} to lowest terms by extracting and canceling out 5.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{200000}{9}-20^{2}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Calculate 20 to the power of 2 and get 400.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{200000}{9}-400}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Subtract 400 from \frac{200000}{9} to get \frac{196400}{9}.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{196400}{9}}{\frac{2\times 10^{-3}}{90\times 10^{-9}}}}$$

Cancel out 2 in both numerator and denominator.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{196400}{9}}{\frac{10^{-3}}{45\times 10^{-9}}}}$$

To divide powers of the same base, subtract the denominator's exponent from the numerator's exponent.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{196400}{9}}{\frac{10^{6}}{45}}}$$

Calculate 10 to the power of 6 and get 1000000.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{196400}{9}}{\frac{1000000}{45}}}$$

Reduce the fraction \frac{1000000}{45} to lowest terms by extracting and canceling out 5.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{\frac{196400}{9}}{\frac{200000}{9}}}$$

Divide \frac{196400}{9} by \frac{200000}{9} by multiplying \frac{196400}{9} by the reciprocal of \frac{200000}{9}.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{196400}{9}\times \frac{9}{200000}}$$

Multiply \frac{196400}{9} and \frac{9}{200000} to get \frac{491}{500}.

$$\frac{50000\sqrt{5}}{3\pi }\sqrt{\frac{491}{500}}$$

Rewrite the square root of the division \sqrt{\frac{491}{500}} as the division of square roots \frac{\sqrt{491}}{\sqrt{500}}.

$$\frac{50000\sqrt{5}}{3\pi }\times \frac{\sqrt{491}}{\sqrt{500}}$$

Factor 500=10^{2}\times 5. Rewrite the square root of the product \sqrt{10^{2}\times 5} as the product of square roots \sqrt{10^{2}}\sqrt{5}. Take the square root of 10^{2}.

$$\frac{50000\sqrt{5}}{3\pi }\times \frac{\sqrt{491}}{10\sqrt{5}}$$

Rationalize the denominator of \frac{\sqrt{491}}{10\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{50000\sqrt{5}}{3\pi }\times \frac{\sqrt{491}\sqrt{5}}{10\left(\sqrt{5}\right)^{2}}$$

The square of \sqrt{5} is 5.

$$\frac{50000\sqrt{5}}{3\pi }\times \frac{\sqrt{491}\sqrt{5}}{10\times 5}$$

To multiply \sqrt{491} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{50000\sqrt{5}}{3\pi }\times \frac{\sqrt{2455}}{10\times 5}$$

Multiply 10 and 5 to get 50.

$$\frac{50000\sqrt{5}}{3\pi }\times \frac{\sqrt{2455}}{50}$$

Multiply \frac{50000\sqrt{5}}{3\pi } times \frac{\sqrt{2455}}{50} by multiplying numerator times numerator and denominator times denominator.

$$\frac{50000\sqrt{5}\sqrt{2455}}{3\pi \times 50}$$

Cancel out 50 in both numerator and denominator.

$$\frac{1000\sqrt{5}\sqrt{2455}}{3\pi }$$

Factor 2455=5\times 491. Rewrite the square root of the product \sqrt{5\times 491} as the product of square roots \sqrt{5}\sqrt{491}.

$$\frac{1000\sqrt{5}\sqrt{5}\sqrt{491}}{3\pi }$$

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

$$\frac{1000\times 5\sqrt{491}}{3\pi }$$

Multiply 1000 and 5 to get 5000.

$$\frac{5000\sqrt{491}}{3\pi }$$