1: \(\sqrt[3]{729}=\sqrt[3]{9^3}=9\)
2: \(\dfrac{\sqrt[3]{-2}}{\sqrt[3]{64}}=\dfrac{\sqrt[3]{-2}}{\sqrt[3]{4^3}}=\dfrac{\sqrt[3]{-2}}{4}\)
4: \(\sqrt[3]{-27}=\sqrt[3]{\left(-3\right)^3}=-3\)
5: \(-\dfrac{\sqrt[3]{4}}{\sqrt[3]{32}}=-\sqrt[3]{\dfrac{4}{32}}=-\sqrt[3]{\dfrac{1}{8}}=-\sqrt[3]{\left(\dfrac{1}{2}\right)^3}=-\dfrac{1}{2}\)
7: \(\sqrt[3]{1331}=\sqrt[3]{11^3}=11\)
8: \(\sqrt[3]{-\dfrac{1}{125}}=\sqrt[3]{\left(-\dfrac{1}{5}\right)^3}=-\dfrac{1}{5}\)
10:\(\sqrt[3]{-125}=\sqrt[3]{\left(-5\right)^3}=-5\)
11: \(\sqrt[3]{\dfrac{125}{27}}=\sqrt[3]{\left(\dfrac{5}{3}\right)^3}=\dfrac{5}{3}\)
3:
\(A=\sqrt[3]{2+\sqrt{5}}+\sqrt[3]{2-\sqrt{5}}\)
=>\(A^3=2+\sqrt{5}+2-\sqrt{5}+3\cdot A\cdot\sqrt[3]{\left(2+\sqrt{5}\right)\left(2-\sqrt{5}\right)}\)
=>\(A^3=4-3A\)
=>\(A^3+3A-4=0\)
=>\(A^3-A+4A-4=0\)
=>\(A\left(A-1\right)\left(A+1\right)+4\left(A-1\right)=0\)
=>\(\left(A-1\right)\left(A^2+A+4\right)=0\)
=>A-1=0
=>A=1
6: \(A=\sqrt[3]{9+4\sqrt{5}}+\sqrt[3]{9-4\sqrt{5}}\)
\(\Leftrightarrow A^3=9+4\sqrt{5}+9-4\sqrt{5}+3\cdot A\cdot\sqrt[3]{\left(9+4\sqrt{5}\right)\left(9-4\sqrt{5}\right)}\)
=>\(A^3=18+3A\)
=>\(A^3-3A-18=0\)
=>\(A^3-9A+6A-18=0\)
=>\(A\left(A-3\right)\left(A+3\right)+6\left(A-3\right)=0\)
=>\(\left(A-3\right)\left(A^2+3A+6\right)=0\)
=>A-3=0
=>A=3
9: \(A=\sqrt[3]{5\sqrt{2}+7}-\sqrt[3]{5\sqrt{2}-7}\)
=>\(A^3=5\sqrt{2}+7-5\sqrt{2}+7-3\cdot A\cdot\sqrt[3]{\left(5\sqrt{2}+7\right)\left(5\sqrt{2}-7\right)}\)
=>\(A^3=14-3A\)
=>\(A^3+3A-14=0\)
=>\(A^3-4A+7A-14=0\)
=>\(A\left(A-2\right)\left(A+2\right)+7\left(A-2\right)=0\)
=>\(\left(A-2\right)\left(A^2+2A+7\right)=0\)
=>A-2=0
=>A=2
