Question

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

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

$x^{1/6}$

$x^{1/4}$

$x^{3/8}$

$x^{1/2}$

$x$

知识点：指数根式

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

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

所以正确答案是 D。

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

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

Thus, the correct answer is D.