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Question Number 208676 by RoseAli last updated on 20/Jun/24

	Answered by Berbere last updated on 20/Jun/24
	$(a)^(1/4) +(b)^(1/4) =(√(4+3(√2)))=(2)^(1/4) .(√(2(√2)+3)) ⇔((2a))^(1/4) +((2b))^(1/4) =(√(((√2)+1)^2 ))=(√2)+1 ⇔((a/2))^(1/4) +((b/2))^(1/4) =(√2)+1 b=0 ⇒((a/2))^(1/4) =(√2)+1 no solution N b=1 no solution b=2⇒a=8 b=3 no solution b=4no solution b=8;a=2 (a,b)∈{(2,8);(8,2)}$
	$\sqrt[4]{a} + \sqrt[4]{b} = \sqrt{4 + 3 \sqrt{2}} = \sqrt[4]{2} . \sqrt{2 \sqrt{2} + 3}$ $\Leftrightarrow \sqrt[4]{2 a} + \sqrt[4]{2 b} = \sqrt{{(\sqrt{2} + 1)}^{2}} = \sqrt{2} + 1$ $\Leftrightarrow \sqrt[4]{\frac{a}{2}} + \sqrt[4]{\frac{b}{2}} = \sqrt{2} + 1$ $b = 0 \Rightarrow \sqrt[4]{\frac{a}{2}} = \sqrt{2} + 1 n o s o l u t i o n N$ $b = 1 n o s o l u t i o n$ $b = 2 \Rightarrow a = 8$ $b = 3 n o s o l u t i o n$ $b = 4 n o s o l u t i o n$ $b = 8; a = 2$ $(a, b) \in {(2, 8); (8, 2)}$
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