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Question Number 582 by ssahoo last updated on 31/Jan/15

Show that (√(5+(√(21)))) +(√(8+(√(55)))) =(√(7+(√(33)))) +(√(6+(√(35))))

$$\mathrm{Show}\:\mathrm{that} \\ $$$$\sqrt{\mathrm{5}+\sqrt{\mathrm{21}}}\:+\sqrt{\mathrm{8}+\sqrt{\mathrm{55}}}\:=\sqrt{\mathrm{7}+\sqrt{\mathrm{33}}}\:+\sqrt{\mathrm{6}+\sqrt{\mathrm{35}}} \\ $$

Commented by prakash jain last updated on 31/Jan/15

(√(5+(√(21))))=a+b a^2 +b^2 +2ab=5+(√(21)) a^2 +b^2 =5 2ab=(√(21)) b=((√(21))/(2a)) a^2 +(((√(21))/(2a)))^2 =5 4a^4 +21−20a^2 =0 4a^4 −14a^2 −6a^2 +21=0 (2a^2 −7)(2a^2 −3)=0 a=((√7)/(√2)), a=((√3)/(√2)) b=((√3)/(√2)), b=((√7)/(√2)) (√(5+(√(21))))=a+b=(((√3)+(√7))/(√2)) Other terms need to be simplified similarly to verify.

$$\sqrt{\mathrm{5}+\sqrt{\mathrm{21}}}={a}+{b} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} +\mathrm{2}{ab}=\mathrm{5}+\sqrt{\mathrm{21}} \\ $$$${a}^{\mathrm{2}} +{b}^{\mathrm{2}} =\mathrm{5} \\ $$$$\mathrm{2}{ab}=\sqrt{\mathrm{21}} \\ $$$${b}=\frac{\sqrt{\mathrm{21}}}{\mathrm{2}{a}} \\ $$$${a}^{\mathrm{2}} +\left(\frac{\sqrt{\mathrm{21}}}{\mathrm{2}{a}}\right)^{\mathrm{2}} =\mathrm{5} \\ $$$$\mathrm{4}{a}^{\mathrm{4}} +\mathrm{21}−\mathrm{20}{a}^{\mathrm{2}} =\mathrm{0} \\ $$$$\mathrm{4}{a}^{\mathrm{4}} −\mathrm{14}{a}^{\mathrm{2}} −\mathrm{6}{a}^{\mathrm{2}} +\mathrm{21}=\mathrm{0} \\ $$$$\left(\mathrm{2}{a}^{\mathrm{2}} −\mathrm{7}\right)\left(\mathrm{2}{a}^{\mathrm{2}} −\mathrm{3}\right)=\mathrm{0} \\ $$$${a}=\frac{\sqrt{\mathrm{7}}}{\sqrt{\mathrm{2}}},\:{a}=\frac{\sqrt{\mathrm{3}}}{\sqrt{\mathrm{2}}} \\ $$$${b}=\frac{\sqrt{\mathrm{3}}}{\sqrt{\mathrm{2}}},\:{b}=\frac{\sqrt{\mathrm{7}}}{\sqrt{\mathrm{2}}} \\ $$$$\sqrt{\mathrm{5}+\sqrt{\mathrm{21}}}={a}+{b}=\frac{\sqrt{\mathrm{3}}+\sqrt{\mathrm{7}}}{\sqrt{\mathrm{2}}} \\ $$$$\mathrm{Other}\:\mathrm{terms}\:\mathrm{need}\:\mathrm{to}\:\mathrm{be}\:\mathrm{simplified}\:\mathrm{similarly} \\ $$$$\mathrm{to}\:\mathrm{verify}. \\ $$

Answered by havandip last updated on 16/Aug/15

(√(5+(√(21))))+(√(8+(√(55))))=(√(7+(√(33))))+(√(6+(√(35)))) (√(5+(√(21))))+(√(8+(√(55))))=(√(7+(√(33))))+(√(6+(√(35))))

$$\sqrt{\mathrm{5}+\sqrt{\mathrm{21}}}+\sqrt{\mathrm{8}+\sqrt{\mathrm{55}}}=\sqrt{\mathrm{7}+\sqrt{\mathrm{33}}}+\sqrt{\mathrm{6}+\sqrt{\mathrm{35}}} \\ $$$$\sqrt{\mathrm{5}+\sqrt{\mathrm{21}}}+\sqrt{\mathrm{8}+\sqrt{\mathrm{55}}}=\sqrt{\mathrm{7}+\sqrt{\mathrm{33}}}+\sqrt{\mathrm{6}+\sqrt{\mathrm{35}}} \\ $$$$ \\ $$

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