2022 AMC 10B 第 5 题

Question

求下式的值：

\frac{\left(1+\frac{1}{3}\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}\text{？}

What is the value of

\frac{\left(1+\frac{1}{3}\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?

\sqrt3

2

\sqrt{15}

4

\sqrt{105}

把每个平方根因子改写成相邻两个分数的乘积。

Rewrite each square-root factor as a product of two nearby fractions

分子中的因子会和根号里的对应因子配对。

Pair each factor in the numerator with the matching factor under the square root

设题中正数表达式为

E

。将它平方，并利用

1-\frac{1}{p^2}=(1-\frac{1}{p})(1+\frac{1}{p})

，可得

\begin{aligned} E^2&=\prod_{p\in\{3,5,7\}}\frac{1+\frac{1}{p}}{1-\frac{1}{p}}\\ &=\frac42\cdot\frac64\cdot\frac86=4 \end{aligned}\text{。}

因为这个表达式为正数，所以

E=2

。

所以正确答案是 B。

Let the given positive expression be

E.

Squaring it and using

1-\frac{1}{p^2}=(1-\frac{1}{p})(1+\frac{1}{p})

gives

\begin{aligned} E^2&=\prod_{p\in\{3,5,7\}}\frac{1+\frac{1}{p}}{1-\frac{1}{p}}\\ &=\frac42\cdot\frac64\cdot\frac86=4. \end{aligned}

Hence

E=2.

Thus, the answer is B .

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