Question

化简 $\sqrt[3]{x\sqrt[3]{x\sqrt[3]{x\sqrt{x}}}}.$

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

$\sqrt{x}$

$\sqrt[3]{x^2}$

$\sqrt[27]{x^2}$

$\sqrt[54]{x}$

$\sqrt[81]{x^{80}}$

知识点：根式指数

从内向外计算， $x\sqrt{x} = x^{3/2}$ ，它的立方根为 $x^{1/2}$ 。

接着 $x \cdot x^{1/2} = x^{3/2}$ ，它的立方根仍为 $x^{1/2}$ 。

于是下一层又是 $x \cdot x^{1/2} = x^{3/2}$ ，开立方后仍为 $x^{1/2} = \sqrt{x}$ 。

所以正确答案是 A。

Working outward, $x\sqrt{x} = x^{3/2},$ and its cube root is $x^{1/2}.$

Then $x \cdot x^{1/2} = x^{3/2},$ whose cube root is again $x^{1/2}.$

Repeating once more, $x \cdot x^{1/2} = x^{3/2},$ whose cube root is $x^{1/2} = \sqrt{x}.$

Thus, the correct answer is A.