2021 AMC 12A Problems/Problem 9

Problem

Which of the following is equivalent to $(2+3)(2^2+3^2)(2^4+3^4)(2^8+3^8)(2^{16}+3^{16})(2^{32}+3^{32})(2^{64}+3^{64})?$ $\textbf{(A)} ~3^{127} + 2^{127} \qquad\textbf{(B)} ~3^{127} + 2^{127} + 2 \cdot 3^{63} + 3 \cdot 2^{63} \qquad\textbf{(C)} ~3^{128}-2^{128} \qquad\textbf{(D)} ~3^{128} + 3^{128} \qquad\textbf{(E)} ~5^{127}$

Solution 1

All you need to do is multiply the entire equation by $(3-2)$ . Then all the terms will easily simplify by difference of squares and you will get $3^{128}-2^{128} or \boxed{C}$ as your final answer. Notice you don't need to worry about $3-2$ because that's equal to $1$ .

Note

See problem 1.

2021 AMC 12A (Problems • Answer Key • Resources)
Preceded by Problem 8	Followed by Problem 10
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All AMC 12 Problems and Solutions

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2021 AMC 12A Problems/Problem 9

Contents

Problem

Solution 1

Note

See also