What is the value of \frac(1+(1)/(3))(1+\frac15)(1+\frac17)√(1-(1)/(3^2))(1-(1)/(5^2))(1-(1)/(7^2))?

2022 AMC 10B Problem 5 — Solution

2022 AMC 10B Problem 5

Attempt Problem 5 of the 2022 AMC 10B below, then check your answer against the professionally curated solution from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2022 AMC 10B solutions, or check the answer key.

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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

Solution:

Let the given positive expression be $E.$ Squaring it and using $1-1/p^2=(1-1/p)(1+1/p)$ gives $\begin{aligned} E^2&=\prod_{p\in\{3,5,7\}}\frac{1+1/p}{1-1/p}\\ &=\frac42\cdot\frac64\cdot\frac86=4. \end{aligned}$ Hence $E=2.$

Thus, the answer is B .

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