Solve sqrt{1+10^-7}-1

$$\sqrt { 1 + 10 ^ { - 7 } } - 1$$

Evaluate

$$\frac{\sqrt{100000010}}{10000}-1\approx 0.00000005$$

Solution Steps

Calculate 10 to the power of -7 and get \frac{1}{10000000}.

$$\sqrt{1+\frac{1}{10000000}}-1$$

Add 1 and \frac{1}{10000000} to get \frac{10000001}{10000000}.

$$\sqrt{\frac{10000001}{10000000}}-1$$

Rewrite the square root of the division \sqrt{\frac{10000001}{10000000}} as the division of square roots \frac{\sqrt{10000001}}{\sqrt{10000000}}.

$$\frac{\sqrt{10000001}}{\sqrt{10000000}}-1$$

Factor 10000000=1000^{2}\times 10. Rewrite the square root of the product \sqrt{1000^{2}\times 10} as the product of square roots \sqrt{1000^{2}}\sqrt{10}. Take the square root of 1000^{2}.

$$\frac{\sqrt{10000001}}{1000\sqrt{10}}-1$$

Rationalize the denominator of \frac{\sqrt{10000001}}{1000\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{\sqrt{10000001}\sqrt{10}}{1000\left(\sqrt{10}\right)^{2}}-1$$

The square of \sqrt{10} is 10.

$$\frac{\sqrt{10000001}\sqrt{10}}{1000\times 10}-1$$

To multiply \sqrt{10000001} and \sqrt{10}, multiply the numbers under the square root.

$$\frac{\sqrt{100000010}}{1000\times 10}-1$$

Multiply 1000 and 10 to get 10000.

$$\frac{\sqrt{100000010}}{10000}-1$$

To add or subtract expressions, expand them to make their denominators the same. Multiply 1 times \frac{10000}{10000}.

$$\frac{\sqrt{100000010}}{10000}-\frac{10000}{10000}$$

Since \frac{\sqrt{100000010}}{10000} and \frac{10000}{10000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{\sqrt{100000010}-10000}{10000}$$