Solve 1/sqrt{1-frac{99999{100000}}}

$$\frac { 1 } { \sqrt { 1 - \frac { 99999 } { 100000 } } }$$

Evaluate

$$\sqrt{100000}\approx 316.227766017$$

Solution Steps

Convert 1 to fraction \frac{100000}{100000}.

$$\frac{1}{\sqrt{\frac{100000}{100000}-\frac{99999}{100000}}}$$

Since \frac{100000}{100000} and \frac{99999}{100000} have the same denominator, subtract them by subtracting their numerators.

$$\frac{1}{\sqrt{\frac{100000-99999}{100000}}}$$

Subtract 99999 from 100000 to get 1.

$$\frac{1}{\sqrt{\frac{1}{100000}}}$$

Rewrite the square root of the division \sqrt{\frac{1}{100000}} as the division of square roots \frac{\sqrt{1}}{\sqrt{100000}}.

$$\frac{1}{\frac{\sqrt{1}}{\sqrt{100000}}}$$

Calculate the square root of 1 and get 1.

$$\frac{1}{\frac{1}{\sqrt{100000}}}$$

Factor 100000=100^{2}\times 10. Rewrite the square root of the product \sqrt{100^{2}\times 10} as the product of square roots \sqrt{100^{2}}\sqrt{10}. Take the square root of 100^{2}.

$$\frac{1}{\frac{1}{100\sqrt{10}}}$$

Rationalize the denominator of \frac{1}{100\sqrt{10}} by multiplying numerator and denominator by \sqrt{10}.

$$\frac{1}{\frac{\sqrt{10}}{100\left(\sqrt{10}\right)^{2}}}$$

The square of \sqrt{10} is 10.

$$\frac{1}{\frac{\sqrt{10}}{100\times 10}}$$

Multiply 100 and 10 to get 1000.

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

Divide 1 by \frac{\sqrt{10}}{1000} by multiplying 1 by the reciprocal of \frac{\sqrt{10}}{1000}.

$$\frac{1000}{\sqrt{10}}$$

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

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

The square of \sqrt{10} is 10.

$$\frac{1000\sqrt{10}}{10}$$

Divide 1000\sqrt{10} by 10 to get 100\sqrt{10}.

$$100\sqrt{10}$$