Question

Упростите и вычислите:
∛(9+√80)+∛(9-√80)=

Answer 1

$9+sqrt{80}=9+4sqrt5=frac18cdot(72+32sqrt5)=frac18cdot(27+27sqrt5+45+5sqrt5)=\=frac18cdot(3^3+3cdot3^2cdotsqrt5+3cdot3cdot(sqrt5)^2+(sqrt5)^3)=frac18cdot(3+sqrt5)^3\\9-sqrt{80}=9-4sqrt5=frac18cdot(72-32sqrt5)=frac18cdot(27-27sqrt5+45-5sqrt5)=\=frac18cdot(3^3-3cdot3^2cdotsqrt5+3cdot3cdot(sqrt5)^2-(sqrt5)^3)=frac18cdot(3-sqrt5)^3\\sqrt[3]{9+sqrt{80}}+sqrt[3]{9-sqrt{80}}=sqrt[3]{frac{(3+sqrt5)^3}8}+sqrt[3]{frac{(3-sqrt5)^3}8}=$
$=frac{3+sqrt5}2+frac{3-sqrt5}2=frac{3+sqrt5+3-sqrt5}2=frac62=3$