2022 AMC 10 B

Complete problem set with solutions and individual problem pages

Problem 5 Easy

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

  • A.

    \sqrt{3}

  • B.

    2

  • C.

    \sqrt{15}

  • D.

    4

  • E.

    \sqrt{105}

Answer:B

Take the difference of squares of the radicand of denominator, we can rewrite the expression in the form of \frac {a}{\sqrt{ab}}, where a=\frac 34 \times \frac 65 \times \frac 87 and b=\frac 23 \times \frac 45 \times \frac 67. Here b=\frac 14a. Substitute and simplify to get the result is 2.

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