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Question Number 212175 by RojaTaniya last updated on 05/Oct/24

Answered by A5T last updated on 05/Oct/24

$$Leta+b+c=x;ab+bc+ca=y;abc=z$$$$a^2+b^2+c^2=x^2-2y=20...\left(i\right)$$$$a^3+b^3+c^3=x(x^2-3y)+3z=x^3-3xy+3z=44...\left(ii\right)$$$$(a+b)(b+c)(c+a)=xy-z=156..\left(iii\right)$$$$\left(ii\right)+3\left(iii\right):x^3=512\Rightarrow x=8\Rightarrow y=22\Rightarrow z=20$$$$\Rightarrow a,b,crootsof:s^3-8s^2+22s-20=0$$$$\Rightarrow (s-2)(s^2-6s+10)=0$$$$\Rightarrow (a,b,c)=(2,3+i,3-i)uptopermutation$$$$\Rightarrow a^6+b^6+c^6=2^6+{(3+i)}^6+{(3-i)}^6=-640$$

Commented by A5T last updated on 05/Oct/24

$$a^6+b^6+c^6={\left(a^2\right)}^3+{\left(b^2\right)}^3+{\left(c^2\right)}^3$$$$=(a^2+b^2+c^2)[{(a^2+b^2+c^2)}^2-3a^2{b}^2-3b^2{c}^2-3c^2{a}^2]+3a^2{b}^2{c}^2$$$$=20[400-3{(ab+bc+ca)}^2+6abc(a+b+c)]+3z^2$$$$=20[400-3{\left(22\right)}^2+3\left(20\right)\left(8\right)]+3{\left(20\right)}^2=-640$$

Commented by A5T last updated on 05/Oct/24

$$a^6+b^6+c^6={\left(a^3\right)}^2+{\left(b^3\right)}^2+{\left(c^3\right)}^2$$$$={(a^3+b^3+c^3)}^2-2(a^3{b}^3+b^3{c}^3+c^3{a}^3)$$$$={44}^2-2[{(ab+bc+ca)}^3-3abc(ab+bc+ca)(a+b+c)$$$$+3{\left(abc\right)}^2]$$$$={44}^2-2[{22}^3-3\left(20\right)\left(22\right)\left(8\right)+3{\left(20\right)}^2]=-640$$

Commented by Ghisom last updated on 05/Oct/24

$$a,b,c\in \mathbb{C}$$$$\Rightarrow$$$$x^3=512\Rightarrow x=8\vee x=-4\pm 4\sqrt{3}\mathrm{i}$$$$\Rightarrow$$$$$$$$[a,b,c\mathrm{ordered}\mathrm{by}\mathrm{angle}]$$$$$$$$\left(1\right)$$$$x=8\wedge y=22\wedge z=20$$$$(s-a)(s+b)(s+c)=$$$$=s^3-8s^2+22s-20=0$$$$a=2$$$$b=3+\mathrm{i}$$$$c=3-\mathrm{i}$$$$a^6+b^6+c^6=-640★$$$$$$$$\left(2\right)$$$$x=-4-4\sqrt{3}\mathrm{i}\wedge y=-26+16\sqrt{3}\mathrm{i}\wedge z=140+40\sqrt{3}\mathrm{i}$$$$(s-a)(s+b)(s+c)=$$$$=s^3+4(1+\sqrt{3}\mathrm{i})s^2-2(13-8\sqrt{3}\mathrm{i})s-20(7+2\sqrt{3}\mathrm{i})=0$$$$a\approx -5.51488+.736675\mathrm{i}$$$$b\approx -2.60820-5.29849\mathrm{i}$$$$c\approx 4.12308-2.36639\mathrm{i}$$$$a^6+b^6+c^6=48320-8640\sqrt{3}\mathrm{i}★$$$$$$$$\left(3\right)$$$$x=-4+4\sqrt{3}\mathrm{i}\wedge y=-26-16\sqrt{3}\mathrm{i}\wedge z=140-40\sqrt{3}\mathrm{i}$$$$(s-a)(s+b)(s+c)=$$$$=s^3+4(1-\sqrt{3}\mathrm{i})s^2-2(13+8\sqrt{3}\mathrm{i})s-20(7-2\sqrt{3}\mathrm{i})=0$$$$a\approx 4.12308+2.36639\mathrm{i}$$$$b\approx -2.60820+5.29849\mathrm{i}$$$$c\approx -5.51488-.736675\mathrm{i}$$$$a^6+b^6+c^6=48320+8640\sqrt{3}\mathrm{i}★$$

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