AMC8 2026

AMC8 2026 · Q9

AMC8 2026 · Q9. It mainly tests Exponents & radicals.

What is the value of this expression? \[\frac{\sqrt{16\sqrt{81}}}{\sqrt{81\sqrt{16}}}\]

这个表达式的值是多少？ \[ \frac{\sqrt{16\sqrt{81}}}{\sqrt{81\sqrt{16}}} \]

(A) \hspace{3pt} \frac{4}{9} \hspace{19pt} \hspace{3pt} \frac{4}{9} \hspace{19pt}

(B) \hspace{3pt} \frac{2}{3} \hspace{19pt} \hspace{3pt} \frac{2}{3} \hspace{19pt}

(D) \hspace{3pt} \frac{3}{2} \hspace{19pt} \hspace{3pt} \frac{3}{2} \hspace{19pt}

(E) \hspace{3pt} \frac{9}{4} \hspace{3pt} \frac{9}{4}

Answer

Correct choice: (B)

正确答案：（B）

Solution

The inner square root cannot be negative. If the outer square root is negative, the denominator is also negative, resulting in a positive fraction, an identity. Thus, we only consider positive squares, in which the numerator simplifies to $\sqrt{16 \cdot 9} = \sqrt{16}\sqrt{9} = 4 \cdot 3 = 12$, and the denominator simplifies to $\sqrt{81 \cdot 4} = \sqrt{81}\sqrt{4} = 9 \cdot 2 = 18$. The answer is $\frac{12}{18} = \frac{4}{6} = \boxed{\frac{2}{3}}$. (Other solutions weren't explicit with logic).

内部的平方根不能为负。如果外部平方根为负，分母也为负，结果是正数分数，恒等式。因此，我们只考虑正平方根，分子化简为$\sqrt{16 \cdot 9} = \sqrt{16}\sqrt{9} = 4 \cdot 3 = 12$，分母化简为$\sqrt{81 \cdot 4} = \sqrt{81}\sqrt{4} = 9 \cdot 2 = 18$。答案为$\frac{12}{18} = \frac{4}{6} = \boxed{\frac{2}{3}}$。（其他解法没有明确逻辑）。

Topics

AL.008 Exponents & radicals