Solve sqrt{(3.7*10^-9)(5.0*10^-10)^2/(8.99*10^9)}

$$\sqrt { \frac { ( 3.7 \times 10 ^ { - 9 } ) ( 5.0 \times 10 ^ { - 10 } ) ^ { 2 } } { ( 8.99 \times 10 ^ { 9 } ) } }$$

Evaluate

$$\frac{\sqrt{332630}}{1798000000000000000000}\approx 3.207679883 \cdot 10^{-19}$$

Solution Steps

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

$$\sqrt{\frac{3.7\times \left(5\times 10^{-10}\right)^{2}}{8.99\times 10^{18}}}$$

Calculate 10 to the power of -10 and get \frac{1}{10000000000}.

$$\sqrt{\frac{3.7\times \left(5\times \frac{1}{10000000000}\right)^{2}}{8.99\times 10^{18}}}$$

Multiply 5 and \frac{1}{10000000000} to get \frac{1}{2000000000}.

$$\sqrt{\frac{3.7\times \left(\frac{1}{2000000000}\right)^{2}}{8.99\times 10^{18}}}$$

Calculate \frac{1}{2000000000} to the power of 2 and get \frac{1}{4000000000000000000}.

$$\sqrt{\frac{3.7\times \frac{1}{4000000000000000000}}{8.99\times 10^{18}}}$$

Multiply 3.7 and \frac{1}{4000000000000000000} to get \frac{37}{40000000000000000000}.

$$\sqrt{\frac{\frac{37}{40000000000000000000}}{8.99\times 10^{18}}}$$

Calculate 10 to the power of 18 and get 1000000000000000000.

$$\sqrt{\frac{\frac{37}{40000000000000000000}}{8.99\times 1000000000000000000}}$$

Multiply 8.99 and 1000000000000000000 to get 8990000000000000000.

$$\sqrt{\frac{\frac{37}{40000000000000000000}}{8990000000000000000}}$$

Express \frac{\frac{37}{40000000000000000000}}{8990000000000000000} as a single fraction.

$$\sqrt{\frac{37}{40000000000000000000\times 8990000000000000000}}$$

Multiply 40000000000000000000 and 8990000000000000000 to get 359600000000000000000000000000000000000.

$$\sqrt{\frac{37}{359600000000000000000000000000000000000}}$$

Rewrite the square root of the division \sqrt{\frac{37}{359600000000000000000000000000000000000}} as the division of square roots \frac{\sqrt{37}}{\sqrt{359600000000000000000000000000000000000}}.

$$\frac{\sqrt{37}}{\sqrt{359600000000000000000000000000000000000}}$$

Factor 359600000000000000000000000000000000000=200000000000000000^{2}\times 8990. Rewrite the square root of the product \sqrt{200000000000000000^{2}\times 8990} as the product of square roots \sqrt{200000000000000000^{2}}\sqrt{8990}. Take the square root of 200000000000000000^{2}.

$$\frac{\sqrt{37}}{200000000000000000\sqrt{8990}}$$

Rationalize the denominator of \frac{\sqrt{37}}{200000000000000000\sqrt{8990}} by multiplying numerator and denominator by \sqrt{8990}.

$$\frac{\sqrt{37}\sqrt{8990}}{200000000000000000\left(\sqrt{8990}\right)^{2}}$$

The square of \sqrt{8990} is 8990.

$$\frac{\sqrt{37}\sqrt{8990}}{200000000000000000\times 8990}$$

To multiply \sqrt{37} and \sqrt{8990}, multiply the numbers under the square root.

$$\frac{\sqrt{332630}}{200000000000000000\times 8990}$$

Multiply 200000000000000000 and 8990 to get 1798000000000000000000.

$$\frac{\sqrt{332630}}{1798000000000000000000}$$