Question

If x = (√2 + 1)/(√2 - 1) and x-y=4√2 then what is the value of x^4+y^4 ?​

Answer 1

$x = \dfrac{ \sqrt{2} + 1 }{ \sqrt{2} - 1 } \times \dfrac{ \sqrt{2} + 1 }{ \sqrt{2} + 1}$

$x = \dfrac{ (\sqrt{2} + 1) ^{2} }{ \sqrt{2 } ^{2} - {1}^{2} }$

$x = \dfrac{2 + 1 + 2 \sqrt{2} }{2 - 1} = 3 + 2 \sqrt{2}$

$x - y = 4 \sqrt{2} \\ y = 3 + 2 \sqrt{2} - 4 \sqrt{2} \\ y = 3 - 2 \sqrt{2}$

${x}^{4} + {y}^{4} \\ = ({x}^{2} ) ^{2} + ( {y}^{2} ) ^{2} \\ =( ( 3 + 2 \sqrt{2} ) ^{2} ) ^{2} + ((3 - 2 \sqrt{2} ) ^{2} ) ^{2} \\ =( 9 + 8 + 12 \sqrt{2} ) ^{2} + (9 + 8 - 12 \sqrt{2} ) ^{2} \\ = (17 + 12 \sqrt{2} ) ^{2} + (17 - 12 \sqrt{2} ) ^{2} \\ = 189 + 288 + 408 \sqrt{2} + 189 + 288- 408 \sqrt{2} \\ = 954$

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