The twist in this question is that you must be sure to answer the question they are actually asking you! The question does not ask you to provide the entire value of the complex number that you get from solving the division problem--it only asks for the real number part of that complex number, which is called . (Please note: Even though  itself is also a real number, as the question states, in the expression  this value  is the coefficient of , so it is part of the imaginary number part of the expression.) Therefore, after you perform the division of complex numbers correctly, you have to focus only on the real number part of your answer, not on the imaginary number part.

But first, you do have to perform the division of complex numbers correctly. The most difficult and critical key step of this process is the very first step: You must multiply both the numerator and denominator by what we call the complex conjugate of the denominator, which is a fancy term that just means you change the sign in the middle of the complex number (before the imaginary part). The reason we do this is that this will make the imaginary part in the denominator disappear! Recall the difference of squares formula in algebra: . This formula works because when you FOIL , the two middle terms  will cancel each other out. Well, the same thing happens here when you multiply the complex number in the denominator by its complex conjugate: the imaginary number middle terms will cancel each other out! So . The middle terms  cancel each other out, leaving us with . Now you have to know that because , therefore . So , and now the negative signs in the second term cancel out, making it positive: . The point of this whole process is that the end result is to simplify the denominator to a single real number, which in this case is . 

To continue the complex number division process, now you must also multiply the original numerator by the complex conjugate of the denominator: . The two middle terms, the imaginary parts, combine: . Continuing, you must know that  as we stated above, so . Finishing the simplification of the numerator, we get .

Conveniently, we now see that our numerator and our denominator  (see above) simplify dramatically! . Wow! (Note: On the ACT, these things never happen by accident. The test question writers carefully design the questions so that they often simplify dramatically like this, when the student solves the questions correctly.) 

Now, by solving the complex number division process correctly, we have rewritten the original expression in the form , which in this case has the value . Recall that you must answer the question they are actually asking you, and the question asks you for the value of . Therefore the correct answer choice is . 

Pro Tip: Experienced math students may be able to save a little time during the solution process: If you know that the question is only asking for the real number part of the final complex number, then when you solve the numerator above, you can skip the imaginary number parts, which are the two middle terms when you FOIL . So you can just solve the terms that will become the real number parts: . Since this is the real number part of the numerator, and the whole denominator is also , now you already know the final real number part will be , which is what the question is actually asking for, so that is the final correct answer choice for this question.

This problem tests your fluency with exponent rules, and gives you a helpful clue to guide you through using them. Here you may see that both 27 and 9 are powers of 3.   and . This allows you to express  as  and  as . Then you can simplify those exponents to get . Since when you divide exponents of the same base you subtract the exponents, you now have , and since  you really have: 

This then tells you that .

Note that had you not immediately seen to express all the numbers in this problem as powers of 3, the fact that the question asks for such a combination of variables, , should be your clue; you're given an exponent problem and asked for a subtraction answer, so that should get you thinking about dividing exponents of the same base to subtract the exponents, and at least give you some fodder for playing with exponent rules until you find a way to make progress.

An important principle of exponents being tested here is that when you multiply/divide exponents of the same base, you add/subtract those exponents. Here you can do the corollary; if you had , you would add together those exponents to get . But in this case you're given the combined exponent  and may want to convert it to  so that you can factor:

 allows you to factor the  terms to get:

You can do the arithmetic to simplify , allowing you to then divide both sides by 3 and have:

So .

This problem hinges on your ability to recognize 16, 4, and 64 all as powers of 4 (or of 2).  If you make that recognition, you can use exponent rules to express the terms as powers of 4:

Since taking one exponent to another means that you multiply the exponents, you can simplify the numerator and have:

And then because when you divide exponents of the same base you can subtract the exponents, you can express this as:

This means that .

You should see on this problem that the numbers used, 2, 4, and 8, are all powers of 2. So to get the exponents in a way to be able to be used together, you can factor each base into a base of 2. That gives you:

Then you can apply the rule that when you take one exponent to another, you multiply the exponents. This then simplifies your equation to:

And now on the left hand side of the equation you can apply another exponent rule, that when you multiply two exponents of the same base, you add the exponents together:

Since the bases here are all the same, you can set the exponents equal. This gives you:

This problem rewards your ability to factor exponents. Here if you factor out common terms in the given equation, you can start to see how the math looks like the correct answer. Factoring negative exponents may feel a bit different from the more traditional factoring that you do more frequently, but the mechanics are the same. Here you can choose to factor out the biggest "number" by sight, , or the number that's technically greatest, . Because all numbers are 2-to-a-power, you'll be factoring out common multiples either way.  

If you factor the common , the expression becomes:

Here you can do the arithmetic on the smaller exponents. They convert to:

When you sum the fractions (and 1) within the parentheses, you get:

And since  you can express this now as:

, which converts to the correct answer: 

Note that you could also have started by factoring out  from the given expression. Had you gone that route, the factorization would have led to:

This also gives you the correct answer, as when you sum the terms within parentheses you end up with: 

Whenever you are given addition or subtraction of two exponential terms with a common base, a good first instinct is to factor the addition or subtraction problem to create multiplication. Most exponent rules deal with multiplication/division and very few deal with addition/subtraction, so if you're stuck on an exponent problem, factoring can be your best friend.

For the equation ,  can be rewritten as , leveraging the rule that when you multiply exponents of the same base, you add the exponents. This allows you to factor the common  term on the left hand side of the equation to yield:

And of course you can simplify the small subtraction problem within parentheses to get:

And you can take even one further step: since everything in the equation is an exponent but that 4, you can express 4 as  to get all the terms to look alike:

Now you need to see that  can be expressed as  or as . So the equation can look like:

You can then divide both sides by  and be left with:

This proves that .

To solve this problem we must recognize that  can be broken down into 

After breaking  into  we see that the  cancel out

This leaves us with 

To solve this problem we must first simplify  into  and further into 

Then we can multiply  to get 

To find  we first cancel out the  on both sides and then divide  by  and get 

To solve this problem we must first subtract  from both sides

Then we square both sides

Add the  to both sides

Divide both sides by