In order to simplify the radical, we will need to pull out common factors of possible perfect squares.

The expression becomes:  

The radical 14 does not have any common factors of perfect squares.

The answer is:  

Simplify by factoring both radicals by perfect squares.

Replace the terms.

Combine like-terms.

The answer is:  

Multiply the integers outside of the radical.

Multiply all the values inside the radicals to combine as one radical.

Rewrite the radical using factors of perfect squares.

The answer is:  

The values of the radicals can be combined my multiplication.

The value of  can be factored by using known perfect squares as factors.

Replace the term.

The answer is:  

Rewrite each radical as common factors using known perfect squares.

Simplify the expression.

The answer is:  

A radical expression which is the th root of a constant is in simplest form if and only if, when the radicand is expressed as the product of prime factors, no factor appears  or more times. Since  is a cube, or third, root, find the prime factorization of 52, and determine whether any prime factor appears three or more times.

52 can be broken down as

and further as

No prime factor appears three or more times, so  is in simplest form.

First, apply the Quotient of Radicals Property to split the radical into a numerator and a denominator:

Since we are dealing with cube, or third, roots, rationalize the denominator by multiplying both halves of the fraction by the least cube-root radical expression that would eliminate the radical in the denominator. To do this, note that 7 is a prime number. Therefore, to get a perfect cube, it is necessary to multiply both halves by , and apply the Product of Radicals Property. The reason for this is made more apparent below:

First, apply the Quotient of Radicals Property to split the radical into a numerator and a denominator:

8 is a perfect cube -  - so the denominator can be simplified:

To simplify the numerator, find the prime factorization of its radicand, 250, and look for any prime factors that appear three times:

5 appears three times, so the numerator can be simplified by way of the Product of Radicals Property:

To rationalize a denominator, multiply all terms by the conjugate. In this case, the denominator is , so its conjugate will be .

So we multiply: .

After simplifying, we get .

The first step to simplifying a problem like this one is to convert all radicals to fractional exponents. Remember the following relationship:

Also keep in mind your exponent rules, especially this one:

Now, let's get started on this problem. First, we change that radical expression into something with fractional exponents instead.

Now we use our exponent rules to simplify the numerator.

Finally, we simplify the entire fraction:

We can leave it like this, but it would be better to write it this way, without negative exponents: