Let $x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}},y = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \

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Let $x = \frac{\sqrt{7} + \sqrt{3}}{\sqrt{7} - \sqrt{3}},y = \frac{\sqrt{7} - \sqrt{3}}{\sqrt{7} + \sqrt{3}}$, then the value of $x^{4} + y^{4} + (x + y)^{4}$ is equal to

Accepted Answer

$x = \frac{5 + \sqrt{21}}{2},y = \frac{5 - \sqrt{21}}{2},x + y = 5,xy = 1$

$$\begin{matrix} x^{4} + y^{4} + (x + y)^{4} & \ = \left( x^{2} + y^{2} \right)^{2} - 2(xy)^{2} + (x + y)^{4} \\ & \ = \left\lbrack (x + y)^{2} - 2xy \right\rbrack^{2} - 2x^{2}y^{2} + (x + y)^{4} \\ & \ = (25 - 2)^{2} - 2 + 5^{4} = 1152 \end{matrix}$$

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