positives and negatives

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Each of the following equations has at least one solution EXCEPT


-2n = (-2)-n

2-n = (-2)n

2n = (-2)-n

(-2)n = -2n

(-2)-n = -2-n


I'm not clear on the answer given in the MGMAT book. Another way to explain would be very helpful.

Thanks!
B

[GMATGuruNY]

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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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.

[Brent@GMATPrepNow]

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.

That's right: (any positive number)� = 1
AND (any negative number)� = 1

Cheers,
Brent

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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

[GMATGuruNY]

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.