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 

To solve this problem we first multiply both sides by  to get rid of the fraction

Then we add  to both sides

We move  to the left side to set the equation equal to . This way we are able to factor the equation as if it was a quadratic.

And now we can factor into

Therefore the value of  is

,  does not exist

,