AMC10 2008 B

AMC10 2008 B · Q3

AMC10 2008 B · Q3. It mainly tests Exponents & radicals.

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

假设 $x$ 是一个正实数。以下哪个与 $\sqrt[3]{x\sqrt{x}}$ 等价？

(A) $x^{1/6}$ $x^{1/6}$

(B) $x^{1/4}$ $x^{1/4}$

(D) $x^{1/2}$ $x^{1/2}$

(E) $x$ $x$

Answer

Correct choice: (D)

正确答案：（D）

Solution

The properties of exponents imply that $$ \sqrt[3]{x\sqrt{x}} = (x \cdot x^{1/2})^{1/3} = (x^{3/2})^{1/3} = x^{1/2}. $$

指数性质表明 $$ \sqrt[3]{x\sqrt{x}} = (x \cdot x^{1/2})^{1/3} = (x^{3/2})^{1/3} = x^{1/2}. $$

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