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 .

Whenever you exponential expressions in both the numerator and denominator of a fraction, your first inclination might be to quickly simplify the expressions by canceling terms out in the numerator and denominator.

However, remember to follow the order of operations: you must simplify the numerator and denominator separately to revolve the exponents raised to exponents problem before you can look to cancel terms in the numerator and denominator.

If you consider the numerator, , you should recognize that, because there is no addition or subtraction within the parentheses, that you can simplify this expression by multiplying each exponent within the parentheses by  to get:

.

Similarly, you can simplify the denominator,  by multiplying each exponent within the parentheses by  to get:

.

You can then recombine the numerator and denominator to get . Notice that you now have simple division. Remember that anytime you want to combine two exponential expressions with a common base that are being divided, you simply need to subtract the exponents. If you do, you get:

, which simplifies to .

Whenever you are asked to simplify an expression with exponents and two different bases, you should immediately look to factor. In this case, you should notice that both  and  are powers of . This means that you can rewrite them as  and  respectively.

Once you do this, the numerator becomes .

As you are raising an exponent to an exponent, you should then recognize that you need to multiply the two exponents in order to simplify to get .

Similarly, the denominator becomes , which you can simplify by multiplying the exponents to get .

Your fraction is therefore . Remember that to divide exponents of the same base, simply subtract the exponents. This gives you: 

, or .

Whenever you see addition or subtraction with algebraic terms, you should only think about combining like terms or factoring. Here you have two of one term  and three of another term  so: 

The difficulty in this problem relates primarily to common mistakes with factoring and exponent rules. If you understand exponent rules and how to combine like terms, you will answer this problem quickly and confidently. You should note that the answer choices do not really help you here – they are traps if you make a mistake with exponent rules! Many algebra problems on the SAT exploit common mistakes relating to certain content areas. For instance, in this example, you can see how easy it would be to accidentally add the exponents or add the bases. If you ever make one of these common mistakes, take note and be sure to avoid it the next time you see a similar problem.

This problem tests your ability to combine exponents algebraically, using both the distributive property and the rule for multiplying exponents with the same base. Here it is also helpful to look at the answers to see what the test maker is looking for. In the answer choices, the maximum number of individual terms is 2, and all terms involve a base 10 (or 100). So you should see that your goal is to rewrite as much as possible of what you're given in terms of 10.

In terms of factors/multiples, a 10 is created any time you can pair a 2 with a 5. So as you take the given expression:

Recognize that if you can pair the two 2s you have with two of the 5s in 5555, you'll be able to consider them 10s. So you can rewrite the given expression as:

From there, you can combine the  and  terms: 

And since those are two bases, multiplied, each taken to the same exponent, they'll combine to. That can be rewritten as , making your expression: 

From here, you'll apply the rule that  and add the exponents from the 10s. That gives you: 

The key to beginning this problem is finding common bases. Since 2, 4, and 8 can all be expressed as powers of 2, you will want to factor the 4 into  and the 8 into  so that  can be rewritten as .

From there, you will employ two core exponent rules. First, when you take an exponent to another, you'll multiply the exponents.

That means that:

 becomes 

and

 becomes 

So your new expression is .

Then, when you're multiplying exponents of the same base, you add the exponents. So you can sum 2 + 8 + 24 to get 34, making your simplified exponent .