x=7+3,xy=4=x4+y4=xy=4=>y=4x=47+3=4(7−3)(7+3)(7−3)=7−3x2+y2=(x+y)2−2xy=(27)2−2(7−3)=28−8=20x4+y4=x2+y22−2x2y2=(20)2−2[(7+3)(7−3)]2=400−2(16)=368

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$x = \sqrt{7} + \sqrt{3}, x y = 4 \Rightarrow x^{4} + y^{4} =$

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Detailed Solution

$\begin{array}{l} x = \sqrt{7} + \sqrt{3}, x y = 4 = x^{4} + y^{4} = \\ x y = 4 => y = \frac{4}{x} = \frac{4}{\sqrt{7} + \sqrt{3}} \\ = \frac{4 (\sqrt{7} - \sqrt{3})}{(\sqrt{7} + \sqrt{3}) (\sqrt{7} - \sqrt{3})} = \sqrt{7} - \sqrt{3} \\ x^{2} + y^{2} = (x + y)^{2} - 2 x y \\ = (2 \sqrt{7})^{2} - 2 (7 - 3) = 28 - 8 = 20 \\ x^{4} + y^{4} = {(x^{2} + y^{2})}^{2} - 2 x^{2} y^{2} \\ = (20)^{2} - 2 [(\sqrt{7} + \sqrt{3}) (\sqrt{7} - \sqrt{3})]^{2} \\ = 400 - 2 (16) = 368 \end{array}$