Solve 1/sqrt{2008-sqrt{200}}

$$\frac { 1 } { \sqrt { 2008 } - \sqrt { 200 } }$$

Evaluate

$$\frac{\sqrt{502}+5\sqrt{2}}{904}\approx 0.032606664$$

Solution Steps

Factor 2008=2^{2}\times 502. Rewrite the square root of the product \sqrt{2^{2}\times 502} as the product of square roots \sqrt{2^{2}}\sqrt{502}. Take the square root of 2^{2}.

$$\frac{1}{2\sqrt{502}-\sqrt{200}}$$

Factor 200=10^{2}\times 2. Rewrite the square root of the product \sqrt{10^{2}\times 2} as the product of square roots \sqrt{10^{2}}\sqrt{2}. Take the square root of 10^{2}.

$$\frac{1}{2\sqrt{502}-10\sqrt{2}}$$

Rationalize the denominator of \frac{1}{2\sqrt{502}-10\sqrt{2}} by multiplying numerator and denominator by 2\sqrt{502}+10\sqrt{2}.

$$\frac{2\sqrt{502}+10\sqrt{2}}{\left(2\sqrt{502}-10\sqrt{2}\right)\left(2\sqrt{502}+10\sqrt{2}\right)}$$

Consider \left(2\sqrt{502}-10\sqrt{2}\right)\left(2\sqrt{502}+10\sqrt{2}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{502}+10\sqrt{2}}{\left(2\sqrt{502}\right)^{2}-\left(-10\sqrt{2}\right)^{2}}$$

Expand \left(2\sqrt{502}\right)^{2}.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2^{2}\left(\sqrt{502}\right)^{2}-\left(-10\sqrt{2}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{502}+10\sqrt{2}}{4\left(\sqrt{502}\right)^{2}-\left(-10\sqrt{2}\right)^{2}}$$

The square of \sqrt{502} is 502.

$$\frac{2\sqrt{502}+10\sqrt{2}}{4\times 502-\left(-10\sqrt{2}\right)^{2}}$$

Multiply 4 and 502 to get 2008.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-\left(-10\sqrt{2}\right)^{2}}$$

Expand \left(-10\sqrt{2}\right)^{2}.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-\left(-10\right)^{2}\left(\sqrt{2}\right)^{2}}$$

Calculate -10 to the power of 2 and get 100.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-100\left(\sqrt{2}\right)^{2}}$$

The square of \sqrt{2} is 2.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-100\times 2}$$

Multiply 100 and 2 to get 200.

$$\frac{2\sqrt{502}+10\sqrt{2}}{2008-200}$$

Subtract 200 from 2008 to get 1808.

$$\frac{2\sqrt{502}+10\sqrt{2}}{1808}$$