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 .

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 for your numerator:

 becomes . And for your denominator:  becomes . So 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:

From there, recognize that when you divide exponents of the same base, you subtract the exponents. This means that you have: 

 which simplifies to , so your answer is .

This problem rewards those who see that roots and exponents are the same operation (roots are "fractional exponents"), and who therefore choose the easier order in which to perform the calculation. The trap here is to have you try to square 27. Not only is that labor-intensive, but once you get to 729 you then have to figure out how to take the cube root of that!

Because you can handle the root and the exponent in either order (were you to express this as a fractional exponent, it would be , which proves that the root and exponent are the same operation), you can take the cube root of 27 first if you want to, which you should know is 3. So at that point, your problem is what is ?" And you of course know the answer: it's 9.

It is important to be able to convert between root notation and exponent notation. The third root of a number (for example,  is the same thing as taking that number to the one-third power .

So when you see that you're taking the third root of , you can read that as  to the  power: 

This then allows you to apply the rule that when you take one exponent to another power, you multiply the powers:

This then means that you can express this as:

With roots, it is important that you are comfortable with factoring and with expressing roots as fractional exponents. A square root, for example, can be expressed as taking that base to the  power. Using that rule, the given expression, , could be expressed using fractional exponents as:

This would allow you to then add the exponents and arrive at:

Since that 2 in the denominator of the exponent translates to "square root," you would have the square root of : 

If you were, instead, to work backward from the answer choices, you would see that answer choice  factors to the given expression. If you start with:

You can express that as:

That in turn will factor to:

The first root then simplifies to , leaving you with: 

Therefore, as you can see, choice  factors directly back to the given expression.

Simplify each of the expressions to determine which satisfies the condition of the problem:

When you're working with absolute values and inequalities, recognize that there are two cases that will satisfy the inequality: either the terms within the absolute value directly satisfy the inequality, or they satisfy the "opposite" (flip the sign and multiply by -1). For example, if , then either  or .

For this problem, here is how the two cases will map out:

1) Satisfy the inequality directly:

 means that 

2) Flip the inequality sign and multiply the numeric value by -1:

, so 

This can feel abstract, so it's helpful to test one number for each case to see if it really works. For case 1, try a value greater than 6, so you might try 7.   because 8 > 7, so you have this right.  For case 2, try a number less than -8, such as -9.   because the absolute value of -8 is 8, and 8 is greater than 7.  

When you're working with inequalities and absolute values, make sure that you cover both cases that would satisfy the inequality.

1) The terms within the absolute value directly satisfy the inequality.  Here that's:

, so  and .

2) The "flip the sign and multiply by -1" case. Here that's:

 (note that the inequality sign is flipped, and the value to the right of that sign is multiplied by -1)

, so