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 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.

This problem emphasizes an important concept about mixing exponents and roots: when you see exponents and roots mixed, be ready to treat roots as fractional exponents. That way all of your terms are in a common form, and you can directly apply exponent rules to all terms.

Here, that means that you'll treat  as . With that, the question then becomes:

Now you can use the exponent rule that  (when you divide exponents of the same base, you subtract the exponents) to turn this into:

While you might now look to immediately convert back to the form of the answer choices (which each use radical signs to represent roots),  is not an option. Which brings up another important point about mixing roots with exponents: generally when you're in exponent form you should stay there as long as possible, as the exponential form tends to provide you with more algebraic flexibility.

Here with  , you can pull out the  (or just ) to break apart the mixed-number exponent. You can rephrase this as , which then allows you to convert to an expression that looks like the answer choices. With the only fractional exponent left, , translating to , you can re-express the entire expression as .

This problem creates confusion for people because they are not comfortable rationalizing the cube root in the denominator. To see how to do this properly, first view the given expression like this:

and then ask what you would need to multiply the denominator by (and thus the numerator so that you're effectively multiplying by ) to make the expression under the root sign a perfect cube in the denominator. In this case, you must multiply top and bottom by  as shown below:

.

To simplify this multi-denominator expression, you should first multiply by one in the form  to eliminate the two denominators in the bottom of the expression. This multiplication results in the following:

Now you must multiply by one again to remove the two roots in the denominator. To do that you must recognize the difference of squares and multiply the top and the bottom by the conjugate of  as shown below:

Therefore, the correct answer is .

All of the exponent rules deal with multiplying, rather than adding, bases. In order to turn this into a multiplication question, we count apples (or chickens, or s, or whatever). How many ’s are there here? Three. This expression can be rewritten as . Now the exponent rules will apply; . The answer is .

To solve this problem we must recognize that  can be broken down into 

After breaking  into  we see that the  terms in numerator and denominator cancel out.

This leaves us with 

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 .