Question

Valoarea expresiei ∛(√5+2) -∛(√5-2) =?

Answer 1

$a = \sqrt[3]{\sqrt5 + 2} \\ b = \sqrt[3]{\sqrt5 - 2} \\ \\ a^3-b^3 = \sqrt5+2-(\sqrt5-2) = 4\\ a\cdot b = \sqrt[3]{(\sqrt5+2)(\sqrt5-2)} = 1 \\ \\a^3-b^3= (a-b)(a^2+ab+b^2)\\ a^3-b^3 = (a-b)\Big((a-b)^2+3ab\Big) \\ \\ 4 = (a-b)\Big((a-b)^2+3\Big) \\ \\ a-b = x \\ \\ \Rightarrow 4 = x(x^2+3) \Rightarrow x^3+3x-4 = 0\\ \\ \Rightarrow x = 1\\ \\ \Rightarrow \sqrt[3]{\sqrt5 + 2} - \sqrt[3]{\sqrt5 - 2} = 1$