When dividing radicals, check the denominator to make sure it can be simplified or that there is a radical present that needs to be fixed. Since there is a radical present, we need to eliminate that radical. To do this, we multiply both top and bottom by . The reason is because we want a whole number in the denominator and multiplying by itself will achieve that. By multiplying itself, it creates a square number which can be reduced to .

With the denominator being , the numerator is .

Final answer is .

To simplify this expression, I would start by simplifying the radical on the numerator. Remember, for every pair of the same number underneath the radical, you can take one out of the radical. Therefore, the numerator simplifies to: . Then, get rid of the negative exponent on the denominator (by placing it in the numerator, you get rid of the negative exponent!): . Now simplify like terms so that you get: .

To simplify radicals, I like to approach each term separately. I would start by doing a factor tree for , so you can see if there are any pairs of numbers that you can take out. factors to , so you can take a  out of the radical. For , there are  pairs of 's, so goes outside of the radical, and one  remains underneath the radical. For , there are  pairs of 's, so you can take  's outside the radical. For , there are  complete pairs of 's so goes on the outside, while one  remains underneath the radical. Now, put those all together to get: .

You are not allowed to have radicals in the denominator, so to simplify this you must multiply everything by  

So: 

However, this is not the last step because you need to simplify to numerator, looking for perfect squares.

So:

Then you need to simplify a little further to:

Which of the following is equivalent to the expression ?

When dealing with radicals in denominators, we want to bring them up to the numerator. To do so, multiply the beginning fraction by a fraction equal to 1 which looks like the beginning radical (look at the red part of the following expression).

Now, instead of canceling the square roots, multiply them out and then simplify.

Rationalize the denominator:

Simplify:

To rationalize the numerator you must multiply by the "conjugate" of the numerator. To find the conjugate simply switch the sign of the expression.

*foil the numerator. The middle terms will cancel each other and the radical cancel each other.

To rationalize the denominator you need to multiply the top and bottom by the square root of 14. Upon doing so, you get:

However you can further simplify this because you know that two times forty-nine is ninety-eight you can simplify this to:

To simplify this radical, first we can simplify what is underneath the radical:

Finally, we take the square root of top and bottom:

The square root of 1 is 1, and the square root of .

In order to simplify this problem we need to multiply the top and bottom terms of the fraction by the radical in the denominator. This is because we always want an integer in the denominator.

Simplify.