If-a-1-a-23-then-a-1-4-1-a-1-4-

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

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

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