How do you simplify (sqrt2-sqrt3)/(sqrt2+sqrt3)?

Nov 6, 2017

See a solution process below:

Explanation:

To eliminate the radicals from the denominator (or to rationalize the denominator) we can multiply the fraction by the appropriate form of $1$. In this case;

$\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}}$

$\frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} - \sqrt{3}} \times \frac{\sqrt{2} - \sqrt{3}}{\sqrt{2} + \sqrt{3}} \implies$

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

$\frac{2 - \sqrt{6} - \sqrt{6} + 3}{2 + 0 - 3} \implies$

$\frac{5 - 2 \sqrt{6}}{- 1} \implies$

$- 5 + 2 \sqrt{6}$

Or

$2 \sqrt{6} - 5$