Evaluate the product

MATHEMATICS · Multi-part · AoPS · Mathematics · Special Factorizations · Special Factorizations

Problem

P = \displaystyle \frac{1}{\sqrt{3}+\sqrt{2}} \times \frac{1}{\sqrt{5}-\sqrt{3}}

, and write the final answer with no irrational number in any denominator. (i) Rationalize the denominator of

\displaystyle \frac{1}{\sqrt{3}+\sqrt{2}}

. (ii) Rationalize the denominator of

\displaystyle \frac{1}{\sqrt{5}-\sqrt{3}}

. (iii) Multiply the simplified fractions obtained in (i) and (ii), and simplify

P

to its simplest form.