2021 AMC 10A Spring Problem 10

Below is the video solution and professionally curated solution for Problem 10 of the 2021 AMC 10A Spring, from LIVE by Po-Shen Loh. You can also try the full timed exam, view all 2021 AMC 10A Spring solutions, or check the answer key.

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Concepts:difference of squares telescoping

Difficulty rating: 1070

10.

Which of the following is equivalent to $(2+3)(2^2+3^2)(2^4+3^4)$ $(2^8+3^8)(2^{16}+3^{16})$ $(2^{32}+3^{32})(2^{64}+3^{64})?$

$3^{127} + 2^{127}$

$3^{127} + 2^{127} + 2 \cdot 3^{63}$ $+ 3 \cdot 2^{63}$

$3^{128} - 2^{128}$

$3^{128} + 2^{128}$

$5^{127}$

Video solution:

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Written solution:

Notice that if we multiply by $3 - 2 = 1,$ we end up with a bunch of difference of squares.

$\begin{gather*} (3 - 2)(2 + 3) = 3^2 - 2^2 \\ (3^2 - 2^2)(2^2 + 3^2) = 3^4 - 2^4\\ \vdots \end{gather*}$ This ends up giving us a final value of $3^{128} - 2^{128}.$

Thus, C is the correct answer.

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