1971 AHSME Problems/Problem 7

Problem

$2^{-(2k+1)}-2^{-(2k-1)}+2^{-2k}$ is equal to

$\textbf{(A) }2^{-2k}\qquad \textbf{(B) }2^{-(2k-1)}\qquad \textbf{(C) }-2^{-(2k+1)}\qquad \textbf{(D) }0\qquad \textbf{(E) }2$

Solution

By using the properties of exponents, we can simplify the given expression as follows to obtain our answer: \begin{align*} 2^{-(2k+1)} - 2^{-(2k-1)} + 2^{-2k} &= 2^{-2k-1} - 2^{-2k+1} + 2^{-2k} \\ &= \frac{2^{-2k}}2 - 2\cdot2^{-2k} + 2^{-2k} \\ &= 2^{-2k}(\frac12 - 2 + 1) \\ &= 2^{-2k}(-\frac12) \\ &= -\frac{2^{-2k}}2 \\ &= -2^{-2k-1} \\ &= \boxed{\textbf{(C) }-2^{-(2k+1)}}. \end{align*}

1971 AHSC (Problems • Answer Key • Resources)
Preceded by Problem 6	Followed by Problem 8
1 • 2 • 3 • 4 • 5 • 6 • 7 • 8 • 9 • 10 • 11 • 12 • 13 • 14 • 15 • 16 • 17 • 18 • 19 • 20 • 21 • 22 • 23 • 24 • 25 • 26 • 27 • 28 • 29 • 30 • 31 • 32 • 33 • 34 • 35
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1971 AHSME Problems/Problem 7

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