Solve sqrt{5*10^-4*10^-2/9*10^9}

$$\sqrt{ \frac{ 5 \times { 10 }^{ -4 } \times { 10 }^{ -2 } }{ 9 \times { 10 }^{ 9 } } }$$

Evaluate

$$\frac{\sqrt{2}}{60000000}\approx 0.000000024$$

Solution Steps

To multiply powers of the same base, add their exponents. Add -4 and -2 to get -6.

$$\sqrt{\frac{5\times 10^{-6}}{9\times 10^{9}}}$$

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

$$\sqrt{\frac{5}{9\times 10^{15}}}$$

Calculate 10 to the power of 15 and get 1000000000000000.

$$\sqrt{\frac{5}{9\times 1000000000000000}}$$

Multiply 9 and 1000000000000000 to get 9000000000000000.

$$\sqrt{\frac{5}{9000000000000000}}$$

Reduce the fraction \frac{5}{9000000000000000} to lowest terms by extracting and canceling out 5.

$$\sqrt{\frac{1}{1800000000000000}}$$

Rewrite the square root of the division \sqrt{\frac{1}{1800000000000000}} as the division of square roots \frac{\sqrt{1}}{\sqrt{1800000000000000}}.

$$\frac{\sqrt{1}}{\sqrt{1800000000000000}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\sqrt{1800000000000000}}$$

Factor 1800000000000000=30000000^{2}\times 2. Rewrite the square root of the product \sqrt{30000000^{2}\times 2} as the product of square roots \sqrt{30000000^{2}}\sqrt{2}. Take the square root of 30000000^{2}.

$$\frac{1}{30000000\sqrt{2}}$$

Rationalize the denominator of \frac{1}{30000000\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{2}}{30000000\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{2}}{30000000\times 2}$$

Multiply 30000000 and 2 to get 60000000.

$$\frac{\sqrt{2}}{60000000}$$