Question

tính giá trị của các biểu thức

a)\(\dfrac{\sqrt[3]{4}+\sqrt[3]{2}+2}{\sqrt[3]{4}+\sqrt[3]{2}+1}\)

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

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

Answer 1

a)\(\dfrac{\sqrt[3]{4}+\sqrt[3]{2}+2}{\sqrt[3]{4}+\sqrt[3]{2}+1}=\dfrac{\sqrt[3]{4}+\sqrt[3]{2}+\sqrt[3]{8}}{\sqrt[3]{4}+\sqrt[3]{2}+1}\)

\(=\dfrac{\sqrt[3]{2}\left(\sqrt[3]{2}+1+\sqrt[3]{4}\right)}{\sqrt[3]{2}+1+\sqrt[3]{4}}=\sqrt[3]{2}\)

b)\(\sqrt{3+\sqrt{3}+\sqrt[3]{10+6\sqrt{3}}}=\sqrt{3+\sqrt{3}+\sqrt[3]{\left(1+\sqrt{3}\right)^3}}\)

\(=\sqrt{3+\sqrt{3}+1+\sqrt{3}}=\sqrt{4+2\sqrt{3}}\)

\(=\sqrt{\left(1+\sqrt{3}\right)^2}=1+\sqrt{3}\)

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