Need clarification on Problem Solving

[phoenix9801]

Would you please use Picking Numbers and/or Straightforward Math to solve this question. Please be simple. (not Algebra). Thanks.


1- If M is an integer such that (-2)^2m = 2^9-m, then M= ?

(Please note for this problem I know the answer is m=3, but I do not understand how (-2)^2m is the same as 2^2m ???? )

[gmatjedi]

i would equate 2m=9-m
3m=9
m=3

to answer your other question, any negative number raised to an even power is positive and any positive number raised to even or odd power is also positive.
(-2)^2m is the same as 2^2m because m is an integer and 2multiplied by m will always be even ensuring a positive number on both sides

[GMATGuruNY]

If you're not good with exponents, remember this rule: when in doubt, write it out.

If m=3, then the equation becomes:

(-2)^(2*3) = 2^(9-3)

which simplifies to:

(-2)^6 = 2^6.

Now write out each side of the equation:

(-2)^6 = (-2)(-2)(-2)(-2)(-2)(-2) = 64
2^6 = 2*2*2*2*2*2 = 64

So the two sides of the equation are equal.

Here's the rule that the problem is testing: when a negative number is raised to an even power, the result is always positive:

(-2)^2 = 4
(-3)^4 = 81
(-942)^108 = something too huge to determine, but we do know that the result will be positive because the exponent is even.