Question Number 168054 by Mastermind last updated on 01/Apr/22
$${Simplify}\:\:\frac{\sqrt{\mathrm{1}+{x}}+\sqrt{{x}}}{\:\sqrt{\mathrm{1}−{x}}+\sqrt{{x}}} \\ $$
Commented by MJS_new last updated on 01/Apr/22
$$\mathrm{what}\:\mathrm{do}\:\mathrm{you}\:\mathrm{want}?\:\mathrm{what}\:\mathrm{is}\:“\mathrm{simple}''? \\ $$$$\mathrm{we}\:\mathrm{can}\:\mathrm{do}\:\mathrm{this}: \\ $$$$\frac{\sqrt{\mathrm{1}+{x}}+\sqrt{{x}}}{\:\sqrt{\mathrm{1}−{x}}+\sqrt{{x}}}=\frac{\sqrt{\mathrm{1}+{x}}+\sqrt{{x}}}{\:\sqrt{\mathrm{1}−{x}}+\sqrt{{x}}}×\frac{\sqrt{\mathrm{1}−{x}}−\sqrt{{x}}}{\:\sqrt{\mathrm{1}−{x}}−\sqrt{{x}}}= \\ $$$$=\frac{\sqrt{\mathrm{1}−{x}^{\mathrm{2}} }+\left(\sqrt{\mathrm{1}−{x}}−\sqrt{\mathrm{1}+{x}}\right)\sqrt{{x}}−{x}}{\mathrm{1}−\mathrm{2}{x}} \\ $$$$\mathrm{but}\:\mathrm{is}\:\mathrm{this}\:“\mathrm{simplified}''? \\ $$
Commented by Mastermind last updated on 02/Apr/22
$${You}\:{did}\:{a}\:{great}\:{job},\:{Thanks}\:{so}\:{much} \\ $$
Commented by Mastermind last updated on 02/Apr/22
$${Thanks}\:{to}\:{everyone}\: \\ $$
Answered by HeferH last updated on 01/Apr/22
$$ \\ $$$$\:\frac{\sqrt{\mathrm{1}\:+\:{x}}\:+\:\sqrt{{x}}}{\:\sqrt{\mathrm{1}\:−\:{x}}\:+\:\sqrt{{x}}}\:\centerdot\:\frac{\:\sqrt{\mathrm{1}\:−\:{x}}\:−\:\sqrt{{x}}}{\:\sqrt{\mathrm{1}\:−\:{x}}\:−\:\sqrt{{x}}}\:\Rightarrow \\ $$$$\:\frac{\left(\sqrt{\mathrm{1}\:+{x}}\right)^{\mathrm{2}} \:−\left(\sqrt{{x}}\right)^{\mathrm{2}} }{\left(\sqrt{\mathrm{1}\:−\:{x}}\right)^{\mathrm{2}} \:−\left(\sqrt{{x}}\right)^{\mathrm{2}} }\:=\:\frac{\mathrm{1}\:+\:{x}\:−\:{x}}{\mathrm{1}\:−\:{x}\:−{x}}\:=\:\frac{\mathrm{1}}{\mathrm{1}−\mathrm{2}{x}} \\ $$$$\: \\ $$$$\: \\ $$
Commented by MJS_new last updated on 01/Apr/22
$$\mathrm{wrong}. \\ $$$$\left(\sqrt{\mathrm{1}+{x}}+\sqrt{{x}}\right)\left(\sqrt{\mathrm{1}−{x}}+\sqrt{{x}}\right)\neq\left(\sqrt{\mathrm{1}+{x}}\right)^{\mathrm{2}} −\left(\sqrt{{x}}\right)^{\mathrm{2}} \\ $$
Commented by HeferH last updated on 01/Apr/22
$$\: \\ $$$$\:{yeah},\:{I}\:{made}\:{a}\:{mistake},\:{sorry} \\ $$$$\:\left({didn}'{t}\:{realize}\right) \\ $$$$\: \\ $$