The number of terms in the expansion of [(a+3b)^2(a-3b)^2]^2 when simplified is:

1950 AMC 12 Problem 16

Attempt Problem 16 of the 1950 AMC 12 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 1950 AMC 12 solutions, or check the answer key.

All problems are used with official legal permission of the Mathematical Association of America (MAA).

16.

The number of terms in the expansion of

\big[(a+3b)^2(a-3b)^2\big]^2

when simplified is:

4

5

6

7

8

Combine

a+3b

and

a-3b

before expanding

The expression simplifies to

(a^2-9b^2)^4

Using the difference of squares,

\begin{aligned} &\big[(a+3b)^2(a-3b)^2\big]^2\\ &=\big[(a^2-9b^2)^2\big]^2\\ &=(a^2-9b^2)^4. \end{aligned}

Its binomial expansion has one nonzero term for each exponent choice

0,1,2,3,4,

so it has five terms.

Thus, the correct answer is B.