To expand the exponent, we multiply the base by whatever the exponent is.

Although we have two variables, we do know that a number raised to a zero power is one. Therefore:

We know that when we expand  we get . We have to remember to apply exponents first in PEMDAS before adding the negative sign. The product is .

We know when we expand  we get . Our answer will be positive because we have even number of negative sign. The product is .

Let's look at all the choices. Some of them can be simplified.

 When we plug in .... This is incorrect.

 With base  raising to any power, the answer is always  but when added with another exponent, it will always be great than  so therefore this answer is wrong. .

 Apply the division rule of exponents. . This is not correct.

 With base  raising to any power, the answer is always  regardless of how many time it is raised.

 Although a big mess, it simplifies to . Everything cancels out leaving us with an answer of  which makes this choice correct.

For all : , this means  is a fraction. Let's say . Let's look at some choices.

 No matter what  is, the answer is always .

 If , then that means .  is less than  so that means  is not correct.

 Since  is small and we are dividing a big number, our answer will come out really small.  So that means  is also not right.

 As we saw in the  example, it seems like the numbers will get smaller but when added, it may be bigger than . By doing the math, we get . This is not bigger than  so this choice is false. 

Finally,  also means . No matter what happens, this answer is always bigger than  . This is our final and correct answer. 

The key factor to keep in mind is  is negative and also when we multiply even amount of negative values, the answer is POSITIVE. So therefore we need to find an exponent that is even. We are applying the power rule of exponents so we just multiply the exponents and keep the base the same.

 This is odd and will generate a negative answer. This isn't correct.

 This is even and will generate a positive answer. Let's check more choices.

This is odd and will generate a negative answer. This isn't correct.

 This is odd and will generate a negative answer. This isn't correct.

This is even and will generate a positive answer. Also, because we know , we know that any number in this range will generate huge numbers with an even exponent.  will always hold true as long as  is given. Therefore  is the correct answer.

All we have to do is plug in .

As we know, if we interchanged the base and exponent, they usually won't generate the same answers. We can test this  idea out and can deduce the bigger base with smaller exponential value is greater than a small base with larger exponential value.

Example:   

The only pairs of exponents that violate this is .  .

Therefore the answer is .

Let's look at all the choices and see which will equal .

 Anything to the zero power is . So this choice is wrong.

 This means that . When we see fractional exponents, the format goes:  where  represents the index of the radical, and  is the exponent raising the base . Clearly,  is not . So this choice is wrong.

 is actually . Remember in PEMDAS, we apply the exponent first followed by the negative sign. So this choice is wrong.

 is just . So this choice is wrong.

Finally,  is actually . When dealing with negative exponents, it's just  where the exponent is just the positive version. Therefore . This is the correct answer.