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Question Number 114676 by ZiYangLee last updated on 20/Sep/20
If a^(1/2) −a^(−(1/2)) =1, prove that a+a^(−1) =3
$$\mathrm{If}\:{a}^{\frac{\mathrm{1}}{\mathrm{2}}} −{a}^{−\frac{\mathrm{1}}{\mathrm{2}}} =\mathrm{1},\:\mathrm{prove}\:\mathrm{that}\:{a}+{a}^{−\mathrm{1}} =\mathrm{3} \\ $$
Answered by Olaf last updated on 20/Sep/20
(√a)+(1/( (√a))) = 1  ⇒ ((√a)+(1/( (√a))))^2  = 1  a+(1/a)+2 = 1 ⇒ a+a^(−1)  = −1 !  But if (√a)−(1/( (√a))) = 1  ⇒ ((√a)−(1/( (√a))))^2  = 1  a+(1/a)−2 = 1  a+a^(−1)  = 3
$$\sqrt{{a}}+\frac{\mathrm{1}}{\:\sqrt{{a}}}\:=\:\mathrm{1} \\ $$$$\Rightarrow\:\left(\sqrt{{a}}+\frac{\mathrm{1}}{\:\sqrt{{a}}}\right)^{\mathrm{2}} \:=\:\mathrm{1} \\ $$$${a}+\frac{\mathrm{1}}{{a}}+\mathrm{2}\:=\:\mathrm{1}\:\Rightarrow\:{a}+{a}^{−\mathrm{1}} \:=\:−\mathrm{1}\:! \\ $$$$\mathrm{But}\:\mathrm{if}\:\sqrt{{a}}−\frac{\mathrm{1}}{\:\sqrt{{a}}}\:=\:\mathrm{1} \\ $$$$\Rightarrow\:\left(\sqrt{{a}}−\frac{\mathrm{1}}{\:\sqrt{{a}}}\right)^{\mathrm{2}} \:=\:\mathrm{1} \\ $$$${a}+\frac{\mathrm{1}}{{a}}−\mathrm{2}\:=\:\mathrm{1} \\ $$$${a}+{a}^{−\mathrm{1}} \:=\:\mathrm{3} \\ $$
Commented by ZiYangLee last updated on 20/Sep/20
oops typo haha sry sir
$$\mathrm{oops}\:\mathrm{typo}\:\mathrm{haha}\:\mathrm{sry}\:\mathrm{sir} \\ $$

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