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Question Number 136614 by bemath last updated on 24/Mar/21

If a+(1/a) = 23 then (a)^(1/4)  + (1/( (a)^(1/4) )) =?

$${If}\:{a}+\frac{\mathrm{1}}{{a}}\:=\:\mathrm{23}\:{then}\:\sqrt[{\mathrm{4}}]{{a}}\:+\:\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{{a}}}\:=? \\ $$

Answered by mr W last updated on 24/Mar/21

a+(1/a)=((√a)+(1/( (√a))))^2 −2=23  (√a)+(1/( (√a)))=5  ((a)^(1/4) +(1/( (a)^(1/4) )))^2 −2=5  (a)^(1/4) +(1/( (a)^(1/4) ))=(√7)

$${a}+\frac{\mathrm{1}}{{a}}=\left(\sqrt{{a}}+\frac{\mathrm{1}}{\:\sqrt{{a}}}\right)^{\mathrm{2}} −\mathrm{2}=\mathrm{23} \\ $$$$\sqrt{{a}}+\frac{\mathrm{1}}{\:\sqrt{{a}}}=\mathrm{5} \\ $$$$\left(\sqrt[{\mathrm{4}}]{{a}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{{a}}}\right)^{\mathrm{2}} −\mathrm{2}=\mathrm{5} \\ $$$$\sqrt[{\mathrm{4}}]{{a}}+\frac{\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{{a}}}=\sqrt{\mathrm{7}} \\ $$

Commented by bemath last updated on 24/Mar/21

 or ± (√7) ?

$$\:{or}\:\pm\:\sqrt{\mathrm{7}}\:? \\ $$

Commented by mr W last updated on 24/Mar/21

no! without minus!  (a)^(1/4) >0 just like (√a)>0

$${no}!\:{without}\:{minus}! \\ $$$$\sqrt[{\mathrm{4}}]{{a}}>\mathrm{0}\:{just}\:{like}\:\sqrt{{a}}>\mathrm{0} \\ $$

Commented by bemath last updated on 24/Mar/21

great sir

$${great}\:{sir} \\ $$

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