Question

a)\(\sqrt{3+\sqrt{3}+\sqrt[3]{10+6\sqrt{3}}}\)

b)\(\dfrac{4+2\sqrt{3}}{\sqrt[3]{10+6\sqrt{3}}}\)

Answer 1

\(\text{a) }\sqrt{3+\sqrt{3}+\sqrt[3]{10+6\sqrt{3}}}\\ =\sqrt{3+\sqrt{3}+\sqrt[3]{3\sqrt{3}+9+3\sqrt{3}+1}}\\ =\sqrt{3+\sqrt{3}+\sqrt[3]{\left(\sqrt{3}+1\right)^3}}\\ =\sqrt{3+\sqrt{3}+\sqrt{3}+1}\\ =\sqrt{3+2\sqrt{3}+1}\\ =\sqrt{\left(\sqrt{3}+1\right)^2}=\sqrt{3}+1\)

\(b\text{) }\dfrac{4+2\sqrt{3}}{\sqrt[3]{10+6\sqrt{3}}}=\dfrac{3+2\sqrt{3}+1}{\sqrt[3]{3\sqrt{3}+9+3\sqrt{3}+1}}\\ =\dfrac{\left(\sqrt{3}+1\right)^2}{\sqrt[3]{\left(\sqrt{3}+1\right)^3}}=\dfrac{\left(\sqrt{3}+1\right)^2}{\sqrt{3}+1}=\sqrt{3}+1\)