Solve sqrt{1.77*10^-5*0.2}

$$\sqrt { 1.77 \times 10 ^ { - 5 } \times 0.2 }$$

Evaluate

$$\frac{\sqrt{354}}{10000}\approx 0.001881489$$

Solution Steps

Calculate 10 to the power of -5 and get \frac{1}{100000}.

$$\sqrt{1.77\times \frac{1}{100000}\times 0.2}$$

Multiply 1.77 and \frac{1}{100000} to get \frac{177}{10000000}.

$$\sqrt{\frac{177}{10000000}\times 0.2}$$

Multiply \frac{177}{10000000} and 0.2 to get \frac{177}{50000000}.

$$\sqrt{\frac{177}{50000000}}$$

Rewrite the square root of the division \sqrt{\frac{177}{50000000}} as the division of square roots \frac{\sqrt{177}}{\sqrt{50000000}}.

$$\frac{\sqrt{177}}{\sqrt{50000000}}$$

Factor 50000000=5000^{2}\times 2. Rewrite the square root of the product \sqrt{5000^{2}\times 2} as the product of square roots \sqrt{5000^{2}}\sqrt{2}. Take the square root of 5000^{2}.

$$\frac{\sqrt{177}}{5000\sqrt{2}}$$

Rationalize the denominator of \frac{\sqrt{177}}{5000\sqrt{2}} by multiplying numerator and denominator by \sqrt{2}.

$$\frac{\sqrt{177}\sqrt{2}}{5000\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{\sqrt{177}\sqrt{2}}{5000\times 2}$$

To multiply \sqrt{177} and \sqrt{2}, multiply the numbers under the square root.

$$\frac{\sqrt{354}}{5000\times 2}$$

Multiply 5000 and 2 to get 10000.

$$\frac{\sqrt{354}}{10000}$$