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: