Solve 1/sqrt{2008-sqrt{2007}}

$$\frac { 1 } { \sqrt { 2008 } - \sqrt { 2007 } }$$

Evaluate

$$2\sqrt{502}+3\sqrt{223}\approx 89.610266574$$

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}.

$$\frac{1}{2\sqrt{502}-\sqrt{2007}}$$

Factor 2007=3^{2}\times 223. Rewrite the square root of the product \sqrt{3^{2}\times 223} as the product of square roots \sqrt{3^{2}}\sqrt{223}. Take the square root of 3^{2}.

$$\frac{1}{2\sqrt{502}-3\sqrt{223}}$$

Rationalize the denominator of \frac{1}{2\sqrt{502}-3\sqrt{223}} by multiplying numerator and denominator by 2\sqrt{502}+3\sqrt{223}.

$$\frac{2\sqrt{502}+3\sqrt{223}}{\left(2\sqrt{502}-3\sqrt{223}\right)\left(2\sqrt{502}+3\sqrt{223}\right)}$$

Consider \left(2\sqrt{502}-3\sqrt{223}\right)\left(2\sqrt{502}+3\sqrt{223}\right). Multiplication can be transformed into difference of squares using the rule: \left(a-b\right)\left(a+b\right)=a^{2}-b^{2}.

$$\frac{2\sqrt{502}+3\sqrt{223}}{\left(2\sqrt{502}\right)^{2}-\left(-3\sqrt{223}\right)^{2}}$$

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

$$\frac{2\sqrt{502}+3\sqrt{223}}{2^{2}\left(\sqrt{502}\right)^{2}-\left(-3\sqrt{223}\right)^{2}}$$

Calculate 2 to the power of 2 and get 4.

$$\frac{2\sqrt{502}+3\sqrt{223}}{4\left(\sqrt{502}\right)^{2}-\left(-3\sqrt{223}\right)^{2}}$$

The square of \sqrt{502} is 502.

$$\frac{2\sqrt{502}+3\sqrt{223}}{4\times 502-\left(-3\sqrt{223}\right)^{2}}$$

Multiply 4 and 502 to get 2008.

$$\frac{2\sqrt{502}+3\sqrt{223}}{2008-\left(-3\sqrt{223}\right)^{2}}$$

Expand \left(-3\sqrt{223}\right)^{2}.

$$\frac{2\sqrt{502}+3\sqrt{223}}{2008-\left(-3\right)^{2}\left(\sqrt{223}\right)^{2}}$$

Calculate -3 to the power of 2 and get 9.

$$\frac{2\sqrt{502}+3\sqrt{223}}{2008-9\left(\sqrt{223}\right)^{2}}$$

The square of \sqrt{223} is 223.

$$\frac{2\sqrt{502}+3\sqrt{223}}{2008-9\times 223}$$

Multiply 9 and 223 to get 2007.

$$\frac{2\sqrt{502}+3\sqrt{223}}{2008-2007}$$

Subtract 2007 from 2008 to get 1.

$$\frac{2\sqrt{502}+3\sqrt{223}}{1}$$

Anything divided by one gives itself.

$$2\sqrt{502}+3\sqrt{223}$$