Question Number 58390 by MJS last updated on 22/Apr/19
$$\mathrm{write}\:\mathrm{without}\:\mathrm{roots}\:\mathrm{in}\:\mathrm{denominator}\:\mathrm{if}\:\mathrm{possible} \\ $$$$\left(\mathrm{1}\right)\:\frac{\mathrm{1}}{\:\sqrt{{a}}} \\ $$$$\left(\mathrm{2}\right)\:\frac{\mathrm{1}}{\:\sqrt{{a}}+\sqrt{{b}}} \\ $$$$\left(\mathrm{3}\right)\:\frac{\mathrm{1}}{\:\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}} \\ $$$$\left(\mathrm{4}\right)\:\frac{\mathrm{1}}{\:\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}+\sqrt{{d}}} \\ $$$$\left(\mathrm{5}\right)\:\frac{\mathrm{1}}{\:\sqrt{{a}}+\sqrt{{b}}+\sqrt{{c}}+\sqrt{{d}}+\sqrt{{e}}} \\ $$
Commented by tanmay last updated on 22/Apr/19
$${sir}\:{pls}\:{check}\:{my}\:{answer}… \\ $$
Commented by MJS last updated on 22/Apr/19
$$\mathrm{your}\:\mathrm{answers}\:\mathrm{are}\:\mathrm{right},\:\mathrm{thank}\:\mathrm{you} \\ $$$$\mathrm{I}\:\mathrm{think}\:\mathrm{the}\:\mathrm{5}^{\mathrm{th}} \:\mathrm{one}\:\mathrm{isn}'\mathrm{t}\:\mathrm{possible}\:\mathrm{but}\:\mathrm{I}'\mathrm{m}\:\mathrm{not} \\ $$$$\mathrm{sure}. \\ $$
Answered by tanmay last updated on 22/Apr/19
$$\left.\mathrm{1}\right)\frac{\sqrt{{a}}}{{a}} \\ $$$$\left.\mathrm{2}\right)\frac{\sqrt{{a}}−\sqrt{{b}}}{{a}−{b}} \\ $$$$\left.\mathrm{3}\right)\frac{\sqrt{{a}}\:+\sqrt{{b}}\:−\sqrt{{c}}}{\left(\sqrt{{a}}\:+\sqrt{{b}}\:\right)^{\mathrm{2}} −{c}} \\ $$$$=\frac{\sqrt{{a}}\:+\sqrt{{b}}\:−\sqrt{{c}}}{{a}+{b}−{c}+\mathrm{2}\sqrt{{a}}\:\sqrt{{b}}}\: \\ $$$$=\frac{\left(\sqrt{{a}}\:+\sqrt{{b}}\:−\sqrt{{c}}\:\right)\left({a}+{b}−{c}−\mathrm{2}\sqrt{{a}}\:\sqrt{{b}}\:\right)}{\left({a}+{b}−{c}\right)^{\mathrm{2}} −\mathrm{4}{ab}} \\ $$
Answered by tanmay last updated on 22/Apr/19
$$\left.\mathrm{4}\right)\frac{\left(\sqrt{{a}}\:+\sqrt{{b}}\:\right)−\left(\sqrt{{c}}\:+\sqrt{{d}}\:\right)}{{a}+{b}−{c}−{d}+\mathrm{2}\sqrt{{a}}\:\sqrt{{b}\:}\:−\mathrm{2}\sqrt{{c}}\:\sqrt{{d}}} \\ $$$$\frac{\left\{\left(\sqrt{{a}}\:+\sqrt{{b}}\:\right)−\left(\sqrt{{c}}\:+\sqrt{{d}}\:\right\}\left\{{a}+{b}−{c}−{d}−\mathrm{2}\sqrt{{a}}\:\sqrt{{b}}\:+\mathrm{2}\sqrt{{cd}}\:\right\}\right.}{\left\{\left({a}+{b}−{c}−{d}\right)^{\mathrm{2}} −\mathrm{4}{ab}−\mathrm{4}{cd}\right\}+\mathrm{8}\sqrt{{abcd}}\:}\: \\ $$$$=\frac{\left\{\left(\sqrt{{a}}\:+\sqrt{{b}}\:\right)−\left(\sqrt{{c}}\:+\sqrt{{d}}\:\right)\right\}\left\{{a}+{b}−{c}−{d}−\mathrm{2}\sqrt{{ab}}\:+\mathrm{2}\sqrt{{cd}}\:\right\}\left[\left\{\left({a}+{b}−{c}−{d}\right)^{\mathrm{2}} −\mathrm{4}{ab}−\mathrm{4}{cd}\right\}−\mathrm{8}\sqrt{{abcd}}\:\right\}}{\left\{\left({a}+{b}−{c}−{d}\right)^{\mathrm{2}} −\mathrm{4}{ab}−\mathrm{4}{cd}\right\}^{\mathrm{2}} −\mathrm{64}{abcd}} \\ $$$$ \\ $$
Commented by malwaan last updated on 23/Apr/19
$${try}\:{to}\:{solve}\:\mathrm{5} \\ $$$${please} \\ $$