With problems like this, it is always best to break apart your values into their prime factors. Let's look at the numerator and the denominator separately:

Numerator

Continuing the simplification:

Now, these factors have in common a . Factor this out:

Denominator

This is much simpler:

Now, return to your fraction:

Cancel out the common factors of :

This problem is quite simple if you recall that the units place of powers of 2 follows a simple 4-step sequence. 

Observe the first few powers of 2:

2¹ = 2, 2² = 4, 2³ = 8, 2⁴ = 16, 2⁵ = 32, 2⁶ = 64, 2⁷ = 128, 2⁸ = 256 . . .

The units place follows a sequence of 2, 4, 8, 6, 2, 4, 8, 6, etc. Thus, divide 102 by 4. This gives a remainder of 2.  

The second number in the sequence is 4, so the answer is 4.

For exponent problems like this, the easiest thing to do is to break down all the numbers that you have into their prime factors. Begin with the number given to you:

Now, in order for you to have a number that is a multiple of this, you will need to have at least  in the prime factorization of the given number.  For each of the answer choices, you have:

; This is the answer.

Because the numbers involved in your fraction are so large, you are going to need to do some careful manipulating to get your answer. (A basic calculator will not work for something like this.) These sorts of questions almost always work well when you isolate the large factors and notice patterns involved. Let's first focus on the numerator. Go ahead and break apart the  into its prime factors:

Note that these have a common factor of . Therefore, you can rewrite the numerator as:

Now, put this back into your fraction:

To simplify the following, a common denominator must be achieved. In this case, the first term must be multiplied by (x+2) in both the numerator and denominator and likewise with the second term with (x-3).

Find the least common denominator between x-3 and x-4, which is (x-3)(x-4). Therefore, you have .  Multiplying the terms out equals . Combining like terms results in .

In order to add fractions, we must first make sure they have the same denominator.

So, we multiply  by  and get the following:

Then, we add across the numerators and simplify:

For binomial expressions, it is often faster to simply FOIL them together to find a common trinomial than it is to look for individual least common denominators. Let's do that here:

FOIL and simplify.

Combine numerators.

Thus, our answer is 

To add the two fractions, a common denominator must be found. With one-term denominators, it is easier to simply find the least common denominator between them and multiply each side to obtain it.

In this case, the least common denominator between  and  is . So the first fraction needs to be multiplied by  and the second by :

Now, we can add straight across, remembering to combine terms where we can.

So, our simplified answer is 

Solve the first equation for .

Solve the second equation for . 

The final step is to multiply  and .