Question

cho \(f\left(x\right)\)=(x\(^3\)+12x-31)\(^{2016}\)

tính \(f\left(a\right)\) với a=\(\sqrt[3]{16-8\sqrt{5}}\)+\(\sqrt[3]{16+8\sqrt{5}}\)

Answer 1

\(a^3=16-8\sqrt{5}+16+8\sqrt{5}+3.\sqrt[3]{16^2-8^2.5}a\)

\(a^3=32+3.\sqrt[3]{4^3\left(4-5\right)}a=32-12a\)

\(f\left(x\right)=\left[\left(32-12a\right)+12a-31\right]^{2016}=1^{2016}=1\)

Answer 2

a=\(\sqrt[3]{16-8\sqrt{5}}\)+\(\sqrt[3]{16+8\sqrt{5}}\)

=\(\sqrt[3]{1-3\sqrt{5}+15-5\sqrt{5}}+\sqrt[3]{1+3\sqrt{5}+15+5\sqrt{5}}\)=\(\sqrt[3]{\left(1-\sqrt{5}\right)^3}+\sqrt[3]{\left(1+\sqrt{5}\right)^3}\)

=1-\(\sqrt{5}+1+\sqrt{5}\)=2

thay vào ta được f(a)=(8+24-31)²⁰¹⁶=(-1)²⁰¹⁶=1