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Question Number 185734 by Shrinava last updated on 26/Jan/23

$$\mathrm{x}=\sqrt{8+\sqrt{40+8\sqrt{5}}}$$$$\mathrm{y}=\sqrt{8-\sqrt{40+8\sqrt{5}}}$$$$\mathrm{Find}:{\mathrm{x}}^6+{\mathrm{y}}^6=?$$

Answered by abdullarasool last updated on 26/Jan/23

Answered by HeferH last updated on 26/Jan/23

$$a=8$$$$b=\sqrt{40+8\sqrt{5}}\Rightarrow$$$$x=\sqrt{a+b}$$$$y=\sqrt{a-b}$$$${\left(x^2\right)}^3+{\left(y^2\right)}^3=(x^2+y^2)(x^4+y^4-x^2{y}^2)$$$$=2a[{(a+b)}^2+{(a-b)}^2-(a-b)(a+b)]$$$$=2a[2a^2+2b^2-a^2+b^2]$$$$=2a(a^2+3b^2)=2a^3+6b^{2}a$$$$\Rightarrow$$$$2\left(8^3\right)+6\cdot {\left(\sqrt{40+8\sqrt{5}}\right)}^{2}8$$$$=1024+48(40+8\sqrt{5})$$$$=1024+1920+384\sqrt{5}$$$$=2944+384\sqrt{5}$$

Answered by mr W last updated on 26/Jan/23

$$x^6+y^6$$$$={(8+\sqrt{40+8\sqrt{5}})}^3+{(8-\sqrt{40+8\sqrt{5}})}^3$$$$={(8+8)}^3-3(8+8)(8+\sqrt{40+8\sqrt{5}}){(8-\sqrt{40+8\sqrt{5}})}^3$$$$=4096-48(64-40-8\sqrt{5})$$$$=2944+384\sqrt{5}$$

Answered by manxsol last updated on 27/Jan/23

$$x=\sqrt{a+\sqrt{b}}$$$$y=\sqrt{a-\sqrt{b}}$$$$x^6+y^6={\left(x^2\right)}^3+{\left(y^2\right)}^3=$$$$[(a+\sqrt{b})+(a-\sqrt{b})[{(a+\sqrt{b})}^2-$$$$(a+\sqrt{b})(a-\sqrt{b})+{(a-\sqrt{b})}^2]=$$$$2a[2(a^2+b)-(a^2-b)]=$$$$2a[a^2+3b]$$$$reemplazando$$$$a=8$$$$b=40+8\sqrt{5}$$$$2944+384\sqrt{5}$$

Commented by Shrinava last updated on 28/Jan/23

$$\mathrm{thank}\mathrm{you}\mathrm{dear}\mathrm{professors}$$

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