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.

This problem involves some creative factoring, and factoring is something you should always look to do whenever you see several exponents amidst some addition or subtraction. You cannot here factor any one thing out of all four terms, but you can group the terms to factor some common elements:

Can become:

And if you factor the common  term you have:

The only real number solution available is , so the correct answer is .

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 .

With this exponent problem, the key to getting the given expression in actionable form is to find common bases. Since both 9 and 27 are powers of 3, you can rewrite the given expression as:

When you've done that, you're ready to apply core exponent rules. When you take one exponent to another, you multiply the exponents. So the numerator becomes  and the denominator becomes . Your new fraction is:

Next deal with the negative exponents, which means that you'll flip each term over the fraction bar and make the exponent positive. This then makes your fraction:

Since when you divide exponents of the same base you subtract their exponents, this simplifies to .

This problem rewards you for being able to factor with exponents. Whenever you're faced with several exponents of the same base separated by addition or subtraction, it is a good idea to factor so that you can get more exponents multiplied together. Here that would mean factoring out the common  term to get:

Now you're in a position to do some arithmetic inside the parentheses, since each of those exponents is one you should recognize or be able to quickly calculate by hand. You have:

Which equals:

Here even if you don't recognize  as , you should look to the answer choices to see lots of 2s with exponents and that may be your clue.  You can simplify this to:

And now you have some options. You might see that with two different bases each taken to the same exponent, you can combine the multiplication to get to  or . Or you might go to the answer choices and eliminate the ones that are close but clearly not correct. Either way, you should find the correct answer, .

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:

While upon first glance it might seem to be a quick problem if you just take the exponent between 61 and 63 and say , a quick test of small numbers should show to you that you can't simply do that. The average of  and , for example, is 5, not . So here you'll have to find a way to leverage the rule that Average = (Sum of Terms)/(Number of Terms).

So, algebraically, the average sets up as , but of course those numbers are far too big to calculate and then add. You can, however, use two clues to your advantage: 1) whenever you're adding or subtracting exponents, it's a good idea to factor (remember, exponents are repetitive multiplication, and factoring creates more multiplication). And 2) the answer choices all have powers of 11 in them with no addition, so you should look to factor out a common 11 term so that your math can look more like the answers. If you do so, you'll find that you have:

And from here, you can actually calculate the numbers in parentheses. That gives you:

If you then finish the math, you'll see that you can sum 121 + 1 to get 122, which divides by 2 to give you 61. So your final answer looks like:

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 .