Question Number 59624 by naka3546 last updated on 12/May/19
$${Rationalize}\:\:\:{the}\:\:{denominator}\:\:{of} \\ $$$$\:\:\:\:\:\:\frac{\mathrm{2}}{\:\sqrt{{x}+\mathrm{2}}\:\:+\:\:\sqrt{{x}+\mathrm{1}}\:\:+\:\:\sqrt{{x}}} \\ $$
Answered by MJS last updated on 12/May/19
$$\frac{\mathrm{1}}{\:\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}}=\frac{\sqrt{{a}}−\sqrt{{b}}−\sqrt{{c}}}{{a}−{b}−{c}−\mathrm{2}\sqrt{{bc}}}= \\ $$$$=\frac{\left(\sqrt{{a}}−\sqrt{{b}}−\sqrt{{c}}\right)\left({a}−{b}−{c}+\mathrm{2}\sqrt{{bc}}\right)}{{a}^{\mathrm{2}} +{b}^{\mathrm{2}} +{c}^{\mathrm{2}} −\mathrm{2}{ab}−\mathrm{2}{ac}−\mathrm{2}{bc}} \\ $$$$\mathrm{now}\:\mathrm{do}\:\mathrm{it}\:\mathrm{yourself} \\ $$