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Question title: "Exponent Inequality": Why can't I simplify the inequality?

Why can't I simplify the inequality like this:

(-1/2 )^ N > -8 
(-1/2) ^ N > -2 ^ 3 
-2 ^ -N > -2 ^ 3 
- N > 3 (equating exponentials with a common base) 
N < -3

Explanation:

What's tricky about this is there are two different things going on --- first of all, the exponent, and what it does to a fraction, and second the +/- sign.

Let's treat the +/- sign first --- (negative thing)^N is negative when (N = odd) and positive when (N = even). Raising negatives to even powers makes them positive. Any positive number is greater than any negative number, so any positive number will be greater than -8, and therefore all even values of N will work because they make the output positive --- even N = 0 (remember, zero is an even number), because (anything)^0 = 1 > -8. So, just from the +/- sign, we know {-10, 0, and 10}, the three even number choices, must be included in the answer.

Now, let's think about what happens to (-1/2) when we raise it to powers.

(-1/2)^1 = -1/2 
(-1/2)^2 = +1/4 
(-1/2)^3 = -1/8

Clearly, when we make N a positive number, N >=1, we get positive and negative fractions all with absolute values less than 1. All of these, whether positive or negative, are to the left of -8 on the number line --- that is to say, any x that is between -1 and 1 must be greater than -8. Therefore, all positive values of N work --- we can add N = 3 to the solution, which is now {-10, 0, 3, 10}.

What happens when we raise (-1/2) to a negative power? Well, you may recall the law of exponents that says a^(-N) = (1/a)^N, or (p/q)^-N = (q/p)^N --- in other words, taking a negative power is the same as raising the reciprocal to the positive power. Well, the reciprocal of 1/2 is 2, and the reciprocal of -1/2 is -2, so

(-1/2)^-1 = (-2)^1 = -2 
(-1/2)^-2 = (-2)^2 = +4 
(-1/2)^-3 = (-2)^3 = -8 
(-1/2)^-4 = (-2)^4 = +16

Clearly, the absolute values are getting bigger and bigger as N gets bigger, and the +/- signs are flip-flopping, - for odd powers and + for even powers, as we already indicated above. All the even powers results in positive outputs, so they would work --- we already took care of that. For the the odd powers --- well, N = -1 has an output of -2, which would work if it were an answer choice, but it's not. N = -3 has an output of -8, and it's not true that -8 is greater than -8 --- the equation -8 = -8 is true, so the inequality -8 > -8 is false --- when the correct relationship is an equality, then any inequality is incorrect. We don't need to figure out the output for N =-7 --- we know the output will be a negative number with an absolute value much bigger than 8, so it will be far to the right of -8 on the number line and thus will not satisfy the inequality. The value N = -3 and N = -7 do not work, so the complete solution is {-10, 0, 3, 10}.

Does that make sense? Let us know if you'd like further clarification or have any other questions and we'd be happy to help! :))

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