2022 AMC 10B Problems/Problem 5

Problem

What is the value of $\frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}}?$ $\textbf{(A)}\ \sqrt3 \qquad\textbf{(B)}\ 2 \qquad\textbf{(C)}\ \sqrt{15} \qquad\textbf{(D)}\ 4 \qquad\textbf{(E)}\ \sqrt{105}$

Solution

We apply the difference of squares to the denominator, and then regroup factors: $\begin{align*} \frac{\left(1+\frac13\right)\left(1+\frac15\right)\left(1+\frac17\right)}{\sqrt{\left(1-\frac{1}{3^2}\right)\left(1-\frac{1}{5^2}\right)\left(1-\frac{1}{7^2}\right)}} &= \frac{1+\frac13}{\sqrt{1+\frac13}}\cdot\frac{1+\frac15}{\sqrt{1+\frac15}}\cdot\frac{1+\frac17}{\sqrt{1+\frac17}}\cdot\frac{1}{\sqrt{1-\frac13}}\cdot\frac{1}{\sqrt{1-\frac15}}\cdot\frac{1}{\sqrt{1-\frac17}} \\ &= \sqrt{1+\frac13}\cdot\sqrt{1+\frac15}\cdot\sqrt{1+\frac17} \end{align*}$

~MRENTHUSIASM

2022 AMC 10B (Problems • Answer Key • Resources)
Preceded by Problem 4	Followed by Problem 6
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2022 AMC 10B Problems/Problem 5

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