Question

simplify 2 root 3 by root 3-root 2 +3 root 2 by root 3 + root 2

Answer 1

Identity used :

$(a + b)(a - b) = {a}^{2} - {b}^{2}$

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

On rationalizing the denominator we get,

$= \frac{2 \sqrt{3} }{ \sqrt{3} - \sqrt{2} } \times \frac{ \sqrt{3} + \sqrt{2} }{ \sqrt{3} + \sqrt{2} } + \frac{3 \sqrt{2} }{ \sqrt{3} + \sqrt{2} } \times \frac{ \sqrt{3} - \sqrt{2} }{ \sqrt{3} - \sqrt{2} } \\ \\ = \frac{2 \sqrt{3}( \sqrt{3} + \sqrt{2} )}{ {( \sqrt{3}) }^{2} - {( \sqrt{2} )}^{2} } + \frac{3 \sqrt{2} ( \sqrt{3} - \sqrt{2}) }{ {( \sqrt{3}) }^{2} - {( \sqrt{2} )}^{2} } \\ \\ = \frac{6 + 2 \sqrt{6} }{3 - 2} + \frac{3 \sqrt{6} - 6 }{3 - 2} \\ \\ = 6 + 2 \sqrt{6} + 3 \sqrt{6} - 6 \\ \\ = 5 \sqrt{6}$