Solve 1/sqrt{1+frac{2{4000}}}

$$\frac{ 1 }{ \sqrt{ 1+ \frac{ 2 }{ 4000 } } }$$

Evaluate

$$\frac{20\sqrt{10005}}{2001}\approx 0.999750094$$

Solution Steps

Reduce the fraction \frac{2}{4000} to lowest terms by extracting and canceling out 2.

$$\frac{1}{\sqrt{1+\frac{1}{2000}}}$$

Convert 1 to fraction \frac{2000}{2000}.

$$\frac{1}{\sqrt{\frac{2000}{2000}+\frac{1}{2000}}}$$

Since \frac{2000}{2000} and \frac{1}{2000} have the same denominator, add them by adding their numerators.

$$\frac{1}{\sqrt{\frac{2000+1}{2000}}}$$

Add 2000 and 1 to get 2001.

$$\frac{1}{\sqrt{\frac{2001}{2000}}}$$

Rewrite the square root of the division \sqrt{\frac{2001}{2000}} as the division of square roots \frac{\sqrt{2001}}{\sqrt{2000}}.

$$\frac{1}{\frac{\sqrt{2001}}{\sqrt{2000}}}$$

Factor 2000=20^{2}\times 5. Rewrite the square root of the product \sqrt{20^{2}\times 5} as the product of square roots \sqrt{20^{2}}\sqrt{5}. Take the square root of 20^{2}.

$$\frac{1}{\frac{\sqrt{2001}}{20\sqrt{5}}}$$

Rationalize the denominator of \frac{\sqrt{2001}}{20\sqrt{5}} by multiplying numerator and denominator by \sqrt{5}.

$$\frac{1}{\frac{\sqrt{2001}\sqrt{5}}{20\left(\sqrt{5}\right)^{2}}}$$

The square of \sqrt{5} is 5.

$$\frac{1}{\frac{\sqrt{2001}\sqrt{5}}{20\times 5}}$$

To multiply \sqrt{2001} and \sqrt{5}, multiply the numbers under the square root.

$$\frac{1}{\frac{\sqrt{10005}}{20\times 5}}$$

Multiply 20 and 5 to get 100.

$$\frac{1}{\frac{\sqrt{10005}}{100}}$$

Divide 1 by \frac{\sqrt{10005}}{100} by multiplying 1 by the reciprocal of \frac{\sqrt{10005}}{100}.

$$\frac{100}{\sqrt{10005}}$$

Rationalize the denominator of \frac{100}{\sqrt{10005}} by multiplying numerator and denominator by \sqrt{10005}.

$$\frac{100\sqrt{10005}}{\left(\sqrt{10005}\right)^{2}}$$

The square of \sqrt{10005} is 10005.

$$\frac{100\sqrt{10005}}{10005}$$

Divide 100\sqrt{10005} by 10005 to get \frac{20}{2001}\sqrt{10005}.

$$\frac{20}{2001}\sqrt{10005}$$