Question

Dacă x=$\sqrt[3]{20+14\sqrt{2} }+\sqrt[3]{20-14\sqrt{2} }$,

să se calculeze valoarea numerică a expresiei E(x)=$x^{4}-6x$.

Answer 1

$x = \sqrt[3]{20+14\sqrt 2}+\sqrt[3]{20-14\sqrt 2} \\ \\ a = \sqrt[3]{20+14\sqrt 2},\quad b = \sqrt[3]{20-14\sqrt 2} \\ \\ x^3 = a^3+b^3+3ab(a+b) \\ \\ x^3 = 40+3\cdot \sqrt[3]{8}\cdot x \\ \\ x^3 -6x=40 \Rightarrow x^3-6x-40 = 0 \\ \\ x_1 = -1\, (F),\,\,\,x_2 = 1\,\,(F),\,x_3 = -2\,(F),\,\,x_3 = 2\,(F),\,\,x_4 = -4\,\,(F), \\ x_5 = 4 \Rightarrow 64-24-40 = 0\,\,(A) \Rightarrow x = 4 \\ \\ \Rightarrow E(x) = 4^4-6\cdot 4 = 256-24 = 232$