To solve for, we must first square both sides to get rid of the radical. We get
We subtract both sides byto get thealone.

We square root both sides to get

Answer choiceandare incorrect.

Answer choiceis incorrect because it was not square rooted.

To solve for, we must first square both sides to get rid of the radical. We get. We subtract both sides byto get thealone.

We divide byto getalone.

We square root both sides to getSinceis not listed as an answer choice, we simplify. The highest square root that can multiply tois. We take theout of the radical to get.

To solve this problem we must first simplify the radical by breaking it up into two partsbecomesthen we simplify intoto get

We multiplyto get, then divide byto get

To solve this problem we must first simplifyintoand further into

Then we can multiplyto get

To findwe first cancel out theon both sides and then dividebyand get

To solve this problem we must first subtractfrom both sides

Then we square both sides

Add theto both sides

Divide both sides by

To solve this problem we first multiply both sides byto get rid of the fraction

Then we addto both sides

We moveto the left 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

Therefore the value ofis

does not exist

To solve this problem we first multiply both sides byto get rid of the fraction

Then we addto both sides

We moveto the left 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

Therefore the value ofis

,does not exist

,

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

We moveto 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 denominator to rationalize the root. If you do so, you will multiplyby

, which is no the same as.

For answer choice, multiplyby.

And since, you can simplify the fraction:

, which matches perfectly. Therefore, answer choiceis 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 betweenand, because theis 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 choicefits 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 findingwith the root equation. There are a few different ways you could solve for:

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

2. Realize that(reverse engineering the root) and see that, somust 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 expressas, 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 expressas just. This then means thatis the correct answer choice.