Question

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Answer 1

$z=sqrt[3]{sqrt3+sqrt2}+sqrt[3]{sqrt3-sqrt2}\\(a+b)^3=a^3+3a^2b+3ab^2+b^3=a^3+b^3+3ab(a+b)\\z^3=(sqrt3+sqrt2)+(sqrt3-sqrt2)+3sqrt[3]{(sqrt3+sqrt2)(sqrt3-sqrt2)}cdot \\cdot (sqrt[3]{sqrt3+sqrt2}+sqrt[3]{sqrt3-sqrt2})=2sqrt3+3sqrt[3]{3-2}cdot \\cdot (sqrt[3]{sqrt3+sqrt2}+sqrt[3]{sqrt3-sqrt2})=2sqrt3+3(sqrt[3]{sqrt3+sqrt2}+sqrt[3]{sqrt3-sqrt2})\\frac{z^3}{3}-z=frac{z^3-3z}{3}=$

$=frac{2sqrt3+3(sqrt[3]{sqrt3+sqrt2}+sqrt[3]{sqrt3-sqrt2})-3(sqrt[3]{sqrt3+sqrt2}+sqrt[3]{sqrt3-sqrt2})}{3}=frac{2sqrt3}{3}$

$2); x=sqrt[3]{sqrt5+2}-sqrt[3]{sqrt5-2}\\(a-b)^3=a^3-b^3-3ab(a-b)\\x^3=(sqrt5+2)-(sqrt5-2)-3sqrt[3]{(sqrt5+2)(sqrt5-2)}cdot \\ cdot (sqrt[3]{ sqrt5+2}-sqrt[3]{sqrt5+2})=4-3sqrt[3]{5-4}(sqrt[3]{sqrt5-2}-sqrt[3]{sqrt5+2})=\\=4-3(sqrt[3]{sqrt5-2}-sqrt[3]{sqrt5+2})\\x^3+3x=4-3(sqrt[3]{sqrt5-2}-sqrt[3]{sqrt5+2})+3(sqrt[3]{sqrt5-2}-sqrt[3]{sqrt5+2})=4$