AMC10 2003 A

AMC10 2003 A · Q9

AMC10 2003 A · Q9. It mainly tests Exponents & radicals.

Simplify $\sqrt[3]{\sqrt[3]{x^3\sqrt[3]{x\sqrt{x}}}}$ .

化简 $\sqrt[3]{\sqrt[3]{x^3\sqrt[3]{x\sqrt{x}}}}$ 。

(A) $\sqrt{x}$ $\sqrt{x}$

(B) $\sqrt[3]{x^2}$ $\sqrt[3]{x^2}$

(D) $\sqrt[3]{54x}$ $\sqrt[3]{54x}$

(E) $\sqrt[3]{81x^{80}}$ $\sqrt[3]{81x^{80}}$

Answer

Correct choice: (A)

正确答案：（A）

Solution

(A) We have $$\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}=\left(x\left(x\left(x\cdot x^{\frac12}\right)^{\frac13}\right)^{\frac13}\right)^{\frac13}$$ $$=\left(x\left(x\left(x^{\frac32}\right)^{\frac13}\right)^{\frac13}\right)^{\frac13}$$ $$=\left(x\left(x\cdot x^{\frac12}\right)^{\frac13}\right)^{\frac13}$$ $$=\left(x\left(x^{\frac32}\right)^{\frac13}\right)^{\frac13}=\left(x\cdot x^{\frac12}\right)^{\frac13}=\left(x^{\frac32}\right)^{\frac13}=x^{\frac12}=\sqrt{x}.$$

(A) 我们有 $$\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}=\left(x\left(x\left(x\cdot x^{\frac12}\right)^{\frac13}\right)^{\frac13}\right)^{\frac13}$$ $$=\left(x\left(x\left(x^{\frac32}\right)^{\frac13}\right)^{\frac13}\right)^{\frac13}$$ $$=\left(x\left(x\cdot x^{\frac12}\right)^{\frac13}\right)^{\frac13}$$ $$=\left(x\left(x^{\frac32}\right)^{\frac13}\right)^{\frac13}=\left(x\cdot x^{\frac12}\right)^{\frac13}=\left(x^{\frac32}\right)^{\frac13}=x^{\frac12}=\sqrt{x}.$$

Topics

AL.008 Exponents & radicals