Solve sqrt{frac{2*(10^-12)/4*1610^2}}{2pi1000}10^10

$$\frac{ \sqrt{ \frac{ 2 \times ( { 10 }^{ -12 } ) }{ 4 \cdot 16 { 10 }^{ 2 } } } }{ 2 \pi 1000 } { 10 }^{ 10 }$$

Evaluate

$$\frac{\sqrt{2}}{16\pi }\approx 0.028134885$$

Solution Steps

Cancel out 2 in both numerator and denominator.

$$\frac{\sqrt{\frac{10^{-12}}{2\times 16\times 10^{2}}}}{2\pi \times 1000}\times 10^{10}$$

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

$$\frac{\sqrt{\frac{1}{2\times 16\times 10^{14}}}}{2\pi \times 1000}\times 10^{10}$$

Multiply 2 and 16 to get 32.

$$\frac{\sqrt{\frac{1}{32\times 10^{14}}}}{2\pi \times 1000}\times 10^{10}$$

Calculate 10 to the power of 14 and get 100000000000000.

$$\frac{\sqrt{\frac{1}{32\times 100000000000000}}}{2\pi \times 1000}\times 10^{10}$$

Multiply 32 and 100000000000000 to get 3200000000000000.

$$\frac{\sqrt{\frac{1}{3200000000000000}}}{2\pi \times 1000}\times 10^{10}$$

Rewrite the square root of the division \sqrt{\frac{1}{3200000000000000}} as the division of square roots \frac{\sqrt{1}}{\sqrt{3200000000000000}}.

$$\frac{\frac{\sqrt{1}}{\sqrt{3200000000000000}}}{2\pi \times 1000}\times 10^{10}$$

Calculate the square root of 1 and get 1.

$$\frac{\frac{1}{\sqrt{3200000000000000}}}{2\pi \times 1000}\times 10^{10}$$

Factor 3200000000000000=40000000^{2}\times 2. Rewrite the square root of the product \sqrt{40000000^{2}\times 2} as the product of square roots \sqrt{40000000^{2}}\sqrt{2}. Take the square root of 40000000^{2}.

$$\frac{\frac{1}{40000000\sqrt{2}}}{2\pi \times 1000}\times 10^{10}$$

Rationalize the denominator of \frac{1}{40000000\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\frac{\sqrt{2}}{40000000\left(\sqrt{2}\right)^{2}}}{2\pi \times 1000}\times 10^{10}$$

The square of \sqrt{2} is 2.

$$\frac{\frac{\sqrt{2}}{40000000\times 2}}{2\pi \times 1000}\times 10^{10}$$

Multiply 40000000 and 2 to get 80000000.

$$\frac{\frac{\sqrt{2}}{80000000}}{2\pi \times 1000}\times 10^{10}$$

Multiply 2 and 1000 to get 2000.

$$\frac{\frac{\sqrt{2}}{80000000}}{2000\pi }\times 10^{10}$$

Express \frac{\frac{\sqrt{2}}{80000000}}{2000\pi } as a single fraction.

$$\frac{\sqrt{2}}{80000000\times 2000\pi }\times 10^{10}$$

Multiply 80000000 and 2000 to get 160000000000.

$$\frac{\sqrt{2}}{160000000000\pi }\times 10^{10}$$

Calculate 10 to the power of 10 and get 10000000000.

$$\frac{\sqrt{2}}{160000000000\pi }\times 10000000000$$

Express \frac{\sqrt{2}}{160000000000\pi }\times 10000000000 as a single fraction.

$$\frac{\sqrt{2}\times 10000000000}{160000000000\pi }$$

Cancel out 10000000000 in both numerator and denominator.

$$\frac{\sqrt{2}}{16\pi }$$