Question

What is the value of $\frac{\left(1+\frac{1}{3}\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?$

$\sqrt3$

$2$

$\sqrt{15}$

$4$

$\sqrt{105}$

Answer: B

Concepts:difference of squares radical

Difficulty rating: 1280

Small Hint:

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

Big Hint:

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

Solution:

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 .

2022 AMC 10B Problem 5

Answer: B

Difficulty rating: 1280

Small Hint:

Big Hint:

Solution:

Problem 5 in Other Years