Convert the given expression into a single radical e.g. the expression inside the radical is:

and the cube root of this is :

Hence  must be equal to 2.

Now

Hence the correct answer is

If we combine into a single logarithmic function we get:

Solving for  we get .

The powers of i form a sequence that repeats every four terms.

i¹ = i

i² = -1

i³ = -i

i⁴ = 1

i⁵ = i

Thus:

i²⁵ = i

i²³ = -i

i²¹ = i

i¹⁹= -i

Now we can evalulate the expression.

i²⁵ - i²³ + i²¹ - i¹⁹ + i¹⁷..... + i 

= i + (-1)(-i) + i + (-1)(i) ..... + i

= i + i + i + i + ..... + i

Each term reduces to +i. Since there are 13 terms in the expression, the final result is 13i.

We know that the expression must be negative. Therefore one or all of the terms x⁷, y⁸ and z¹⁰ must be negative; however, even powers always produce positive numbers, so y⁸ and z¹⁰ will both be positive. Odd powers can produce both negative and positive numbers, depending on whether the base term is negative or positive. In this case, x⁷ must be negative, so x must be negative. Thus, the answer is x < 0.

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: