To solve this problem we must first subtract a square from both sides 

We move  to the right 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

If you try to simplify the expression given in the question, you will have a hard time…it is already simplified! However, if you look at the four answer choices you will realize that most of these contain roots in the denominator. Whenever you see a root in the denominator, you should look to rationalize that denominator. This means that you will multiply the expression by one to get rid of the root.

Consider each answer choice as you attempt to simplify each.

For choice , the expression is already simplified and is not the same. At this point, your time is better spent simplifying those that need it to see if those simplified forms match.

For choice , employ the "multiply by one" strategy of multiplying by the same numerator as the denominator to rationalize the root. If you do so, you will multiply  by 

, which is no the same as .

For answer choice , multiply  by .

And since , you can simplify the fraction:

, which matches perfectly. Therefore, answer choice  is correct.

NOTE: If you want to shortcut the algebra, this problem offers you that opportunity by leveraging the answer choices along with an estimate. You can estimate that the given expression, , is between  and , because the  is between  (which is ) and  (which is ). Therefore you know you are looking for a proper fraction, a fraction in which the numerator is smaller than the denominator. Well, look at your answer choices and you will see that only answer choice  fits that description. So without even doing the math, you can rely on a quick estimate and know that you are correct.

The key to this problem is to avoid mistakes in finding  with the root equation. There are a few different ways you could solve for :

1. Leverage the fact that  and apply that to . That means that . Divide both sides by  and see that , so .

2. Realize that  (reverse engineering the root) and see that , so  must equal .

However you find , you must then apply that value to the exponent expression in the second equation. Now you have . And since you're dealing with exponents, you will want to express  as , meaning that you now have: 

Here you should deal with the negative exponents, the rule for which is that . So the fraction you're given, , can then be transformed to .

Now you have:

Employing another rule of exponents, that of dividing exponents of the same base, you can transform the left-hand side to:

Since you now have everything with a base of , you can express  as just . This then means that  is the correct answer choice.

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 ACT 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 .

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 operations (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 operations), 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.