Question

假设 $x$ 是正实数。下列哪一项等价于 $\sqrt[3]{x\sqrt{x}}\,\text{？}$

Assume that $x$ is a positive real number. Which is equivalent to $\sqrt[3]{x\sqrt{x}}\,?$

$x^{\frac{1}{6}}$

$x^{\frac{1}{4}}$

$x^{\frac{3}{8}}$

$x^{\frac{1}{2}}$

$x$

知识点：指数根式

写成 $\sqrt{x}=x^{\frac{1}{2}}$ 。

Write $\sqrt{x}=x^{\frac{1}{2}}$

此时 $x\sqrt{x}=x^{\frac{3}{2}}$ ，取立方根会把指数除以 $3$ 。

Then $x\sqrt{x}=x^{\frac{3}{2}},$ and the cube root divides the exponent by $3$

因为 $\sqrt{x}=x^{\frac{1}{2}}$ ，所以 $x\sqrt{x}=x^{1}\cdot x^{\frac{1}{2}}=x^{\frac{3}{2}}$ 。

取立方根相当于指数乘以 $\tfrac13$ ，得到 $\left(x^{\frac{3}{2}}\right)^{\frac{1}{3}}=x^{\frac{1}{2}}$ 。

所以正确答案是 D。

Since $\sqrt{x}=x^{\frac{1}{2}},$ we have $x\sqrt{x}=x^{1}\cdot x^{\frac{1}{2}}=x^{\frac{3}{2}}.$

Taking the cube root multiplies the exponent by $\tfrac13,$ giving $\left(x^{\frac{3}{2}}\right)^{\frac{1}{3}}=x^{\frac{1}{2}}.$

Thus, the correct answer is D.