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-

(A) The left side is always negative. Notice that in –2n, you apply the exponent first, then the negative sign. 2 raised to any power is always positive, so the negative sign makes the whole left side negative.

The right side is positive for even values of n and negative for odd values of n. Therefore, the two sides of this equation are reciprocals when n is odd, and opposite reciprocals when n is even; the absolute values won’t be the same unless n = 0 (in which case the left side is –1 and the right side is 1). The signs won’t be the same unless n is odd. Therefore, the equation has no solution. Try a few numbers to see the patterns at work.

(B) The left side is always positive, while the right side is positive for even values of n and negative for odd values of n. Therefore, the two sides of the equation are reciprocals when n is even, and opposite reciprocals when n is odd. The only solution to the equation is n = 0, which produces 1 on both sides.

(C) The left side is always positive, while the right side is positive for even values of n and negative for odd values of n. Therefore, the two sides of the equation are reciprocals when n is even, and opposite reciprocals when n is odd. The only solution to the equation is n = 0, which produces 1 on both sides.

(D) The left side is positive for even values of n and negative for odd values of n, while the right side is always negative; the absolute values of the two sides are always the same (= 2n). Therefore, any odd value of n will solve this equation.

(E) The left side is positive for even values of n and negative for odd values of n, while the right side is always negative; the absolute values of the two sides are always the same (= 2-n). Therefore, any odd value of n will solve this equation.

What's your doubt?

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HTH

Please see this thread: each-of-the-following-equations-has-at-least-one-solution-t8182.html

I wrote an explanation about halfway down--hope it helps you! :-)

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Jamie Nelson
ManhattanGMAT Instructor