positives and negatives

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.

Thank-you!

I get it now when using the number 1, but I am confused on the "0" being used to plug in for "n" on a, d, e. I thought anything to the power "0" is automatically 1? Even if it's a negative.

Can you show me the answer choices with all of the numbers using n=1? I understand anything to the power of 0 = 1, but I don't understand a and c answer choices clearly and I thought I did. I just want to ensure this is 100% clear.

Thanks,
Bethany

[email protected] wrote:Can you show me the answer choices with all of the numbers using n=1? I understand anything to the power of 0 = 1, but I don't understand a and c answer choices clearly and I thought I did. I just want to ensure this is 100% clear.

Thanks,
Bethany

If n=1 is plugged into the answer choices, we get:

A. –(2^n) = (–2)^-n
-(2^1) = (-2)^(-1)
-2 = -1/2.

B. 2^-n = (–2)^n
2^(-1) = (-2)^1
1/2 = -2.

C. 2^n = (–2)^-n
2^1 = (-2)^(-1)
2 = -1/2.

D. (–2)^n = –(2^n)
(-2)^1 = -(2^1)
-2 = -2.

E. (–2)^-n = –(2^-n)
(-2)^(-1) = -(2^-1)
-1/2 = -1/2.