Solve sqrt{20*10^-6*0.02*3+400*10^-12*10000}

$$\sqrt{ 20 \times { 10 }^{ -6 } \times 0.02 \times 3+400 \times { 10 }^{ -12 } \times 10000 }$$

Evaluate

$$\frac{\sqrt{130}}{5000}\approx 0.002280351$$

Solution Steps

Calculate 10 to the power of -6 and get \frac{1}{1000000}.

$$\sqrt{20\times \frac{1}{1000000}\times 0.02\times 3+400\times 10^{-12}\times 10000}$$

Multiply 20 and \frac{1}{1000000} to get \frac{1}{50000}.

$$\sqrt{\frac{1}{50000}\times 0.02\times 3+400\times 10^{-12}\times 10000}$$

Multiply \frac{1}{50000} and 0.02 to get \frac{1}{2500000}.

$$\sqrt{\frac{1}{2500000}\times 3+400\times 10^{-12}\times 10000}$$

Multiply \frac{1}{2500000} and 3 to get \frac{3}{2500000}.

$$\sqrt{\frac{3}{2500000}+400\times 10^{-12}\times 10000}$$

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

$$\sqrt{\frac{3}{2500000}+400\times \frac{1}{1000000000000}\times 10000}$$

Multiply 400 and \frac{1}{1000000000000} to get \frac{1}{2500000000}.

$$\sqrt{\frac{3}{2500000}+\frac{1}{2500000000}\times 10000}$$

Multiply \frac{1}{2500000000} and 10000 to get \frac{1}{250000}.

$$\sqrt{\frac{3}{2500000}+\frac{1}{250000}}$$

Add \frac{3}{2500000} and \frac{1}{250000} to get \frac{13}{2500000}.

$$\sqrt{\frac{13}{2500000}}$$

Rewrite the square root of the division \sqrt{\frac{13}{2500000}} as the division of square roots \frac{\sqrt{13}}{\sqrt{2500000}}.

$$\frac{\sqrt{13}}{\sqrt{2500000}}$$

Factor 2500000=500^{2}\times 10. Rewrite the square root of the product \sqrt{500^{2}\times 10} as the product of square roots \sqrt{500^{2}}\sqrt{10}. Take the square root of 500^{2}.

$$\frac{\sqrt{13}}{500\sqrt{10}}$$

Rationalize the denominator of \frac{\sqrt{13}}{500\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{13}\sqrt{10}}{500\left(\sqrt{10}\right)^{2}}$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{13}\sqrt{10}}{500\times 10}$$

To multiply \sqrt{13} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\sqrt{130}}{500\times 10}$$

Multiply 500 and 10 to get 5000.

$$\frac{\sqrt{130}}{5000}$$