The problem should read as follows:
Each of the following equations has at least one solution EXCEPT:
A. -(2^n) = (-2)^-n
B. 2^-n = (-2)^n
C. 2^n = (-2)^-n
D. (-2)^n = -(2^n)
E. (-2)^-n = -(2^-n)
For many test-takers, the most practical approach will be to eliminate the four answer choices that DO have at least one solution.
Values likely to work in a majority of the answer choices are n=0 and n=1.
Plug n=0 and n=1 into the answer choices:
A. -(2^n) = (-2)^-n
n=0:
-(2^0) = (-2)^-0
-1 = 1. Doesn't work.
n=1:
-(2^1) = (-2)^-1
-2 = -1/2. Doesn't work.
Hold onto A.
B. 2^-n = (-2)^n
n=0:
2^-0 = (-2)^0
1=1.
n=0 is a solution. Eliminate B.
C. 2^n = (-2)^-n
n=0:
2^0 = (-2)^-0
1=1.
n=0 is a solution. Eliminate C.
D. (-2)^n = -(2^n)
n=1:
(-2)^1 = -(2^1)
-2 = -2.
n=1 is a solution. Eliminate D.
E. (-2)^-n = -(2^-n)
n=1:
(-2)^(-1) = -(2^-1)
-1/2 = -1/2
n=1 is a solution. Eliminate E.
The correct answer is
A.
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