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Question Number 225323 by hardmath last updated on 21/Oct/25
Find: (((3 + 2 (5)^(1/4) )/(3 - 2 (5)^(1/4) )))^(1/4) . (((5)^(1/4) - 1)/( (5)^(1/4) + 1)) = ?
$$\mathrm{Find}:\:\:\:\left(\frac{\mathrm{3}\:+\:\mathrm{2}\:\sqrt[{\mathrm{4}}]{\mathrm{5}}}{\mathrm{3}\:-\:\mathrm{2}\:\sqrt[{\mathrm{4}}]{\mathrm{5}}}\right)^{\frac{\mathrm{1}}{\mathrm{4}}} .\:\:\:\frac{\sqrt[{\mathrm{4}}]{\mathrm{5}}\:-\:\mathrm{1}}{\:\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1}}\:\:=\:? \\ $$
Answered by Raphael254 last updated on 21/Oct/25
(((3+2(5)^(1/4) )/(3−2(5)^(1/4) )))^(1/4) ×(((6(√5) + 6 − 4((125))^(1/4) − 4(5)^(1/4) )/(6(√5) + 6 + 4((125))^(1/4) + 4(5)^(1/4) )))^(1/4) = (((18(√5) + 18 − 12((125))^(1/4) − 12(5)^(1/4) + 12((125))^(1/4) + 12(5)^(1/4) − 8((625))^(1/4) − 8(√5))/(18(√5) + 18 + 12((125))^(1/4) + 12(5)^(1/4) − 12((125))^(1/4) − 12(5)^(1/4) − 8((625))^(1/4) − 8(√5))))^(1/4) = (((18(√5) − 8(√5) + 18 − 8×5)/(18(√5) − 8(√5) + 18 − 8×5)))^(1/4) = (((10(√5) − 22)/(10(√5) − 22)))^(1/4) = (1)^(1/4) = 1 ((5)^(1/4) − 1)^4 = ((√5) − 2(5)^(1/4) + 1)((√5) − 2(5)^(1/4) + 1) = 5 − 2((125))^(1/4) + (√5) − 2((125))^(1/4) + 4(√5) − 2(5)^(1/4) + (√5) − 2(5)^(1/4) + 1 = −4((125))^(1/4) − 4(5)^(1/4) + 6(√5) + 6 ((5)^(1/4) + 1)^4 = ((√5) + 2(5)^(1/4) + 1)((√5) + 2(5)^(1/4) + 1) = 5 + 2((125))^(1/4) + (√5) + 2((125))^(1/4) + 4(√5) + 2(5)^(1/4) + (√5) + 2(5)^(1/4) + 1 = 4((125))^(1/4) + 4(5)^(1/4) + 6(√5) + 6
$$ \\ $$$$\sqrt[{\mathrm{4}}]{\frac{\mathrm{3}+\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}}{\mathrm{3}−\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}}}×\sqrt[{\mathrm{4}}]{\frac{\mathrm{6}\sqrt{\mathrm{5}}\:+\:\mathrm{6}\:−\:\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:−\:\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{5}}}{\mathrm{6}\sqrt{\mathrm{5}}\:+\:\mathrm{6}\:+\:\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:+\:\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{5}}}} \\ $$$$ \\ $$$$=\:\sqrt[{\mathrm{4}}]{\frac{\mathrm{18}\sqrt{\mathrm{5}}\:+\:\mathrm{18}\:−\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{125}}}\:−\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{5}}}\:+\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{125}}}\:+\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{5}}}\:−\:\mathrm{8}\sqrt[{\mathrm{4}}]{\mathrm{625}}\:−\:\mathrm{8}\sqrt{\mathrm{5}}}{\mathrm{18}\sqrt{\mathrm{5}}\:+\:\mathrm{18}\:+\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{125}}}\:+\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{5}}}\:−\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{125}}}\:−\:\cancel{\mathrm{12}\sqrt[{\mathrm{4}}]{\mathrm{5}}}\:−\:\mathrm{8}\sqrt[{\mathrm{4}}]{\mathrm{625}}\:−\:\mathrm{8}\sqrt{\mathrm{5}}}} \\ $$$$=\:\sqrt[{\mathrm{4}}]{\frac{\mathrm{18}\sqrt{\mathrm{5}}\:−\:\mathrm{8}\sqrt{\mathrm{5}}\:+\:\mathrm{18}\:−\:\mathrm{8}×\mathrm{5}}{\mathrm{18}\sqrt{\mathrm{5}}\:−\:\mathrm{8}\sqrt{\mathrm{5}}\:+\:\mathrm{18}\:−\:\mathrm{8}×\mathrm{5}}} \\ $$$$=\:\sqrt[{\mathrm{4}}]{\frac{\mathrm{10}\sqrt{\mathrm{5}}\:−\:\mathrm{22}}{\mathrm{10}\sqrt{\mathrm{5}}\:−\:\mathrm{22}}}\:=\:\sqrt[{\mathrm{4}}]{\mathrm{1}}\:=\:\mathrm{1} \\ $$$$ \\ $$$$\left(\sqrt[{\mathrm{4}}]{\mathrm{5}}\:−\:\mathrm{1}\right)^{\mathrm{4}} \:=\:\left(\sqrt{\mathrm{5}}\:−\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1}\right)\left(\sqrt{\mathrm{5}}\:−\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1}\right) \\ $$$$=\:\mathrm{5}\:−\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:+\:\sqrt{\mathrm{5}}\:−\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:+\:\mathrm{4}\sqrt{\mathrm{5}}\:−\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\sqrt{\mathrm{5}}\:−\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1} \\ $$$$=\:−\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:−\:\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{6}\sqrt{\mathrm{5}}\:+\:\mathrm{6} \\ $$$$ \\ $$$$\left(\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1}\right)^{\mathrm{4}} \:=\:\left(\sqrt{\mathrm{5}}\:+\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1}\right)\left(\sqrt{\mathrm{5}}\:+\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1}\right) \\ $$$$=\:\mathrm{5}\:+\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:+\:\sqrt{\mathrm{5}}\:+\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:+\:\mathrm{4}\sqrt{\mathrm{5}}\:+\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\sqrt{\mathrm{5}}\:+\:\mathrm{2}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{1} \\ $$$$=\:\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{125}}\:+\:\mathrm{4}\sqrt[{\mathrm{4}}]{\mathrm{5}}\:+\:\mathrm{6}\sqrt{\mathrm{5}}\:+\:\mathrm{6} \\ $$

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