If n\heartsuit m=n^3m^2, what is (2\heartsuit 4)/(4\heartsuit 2)?

2016 AMC 10B Problem 2 — Solution

2016 AMC 10B Problem 2

Below is the professionally curated solution for Problem 2 of the 2016 AMC 10B, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2016 AMC 10B solutions, or check the answer key.

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Concepts:custom operation exponent

Difficulty rating: 870

If $n\heartsuit m=n^3m^2,$ what is $\frac{2\heartsuit 4}{4\heartsuit 2}?$

$\ \dfrac{1}{4}$

$\ \dfrac{1}{2}$

$\ 1$

$\ 2$

$\ 4$

Solution:

For any nonzero $a,b,$ we have $\dfrac{a\heartsuit b}{b\heartsuit a} = \dfrac{a^3b^2}{b^3 a^2} = \dfrac ab .$ Using $a=2,b=4,$ we get that $\frac{ 2}{4} = \frac 12.$

Thus, the correct answer is B .

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