Simplifying radicals

There are many different types of expressions in algebra that we need to be able to work with in order to simplify or solve equations. Radicals are one such example.

The most common radicals we are likely to come across are square roots. Simplifying square root radicals involves using the following property of radicals:

For the above to be true, x and y must both be non-negative numbers. Otherwise, the solution would involve imaginary numbers, since we cannot take the square root of a negative number.

Using this property, simplifying radicals involves expressing the value under the radical as an equivalent product that includes at least one perfect square. Doing so allows us to remove the square root symbol for the perfect square and replace it with its square root, continuing the process until we cannot use the property to simplify the radical any further.


1. Simplify :

We can't simplify any further because 11 is not a perfect square, so is in simplest form.

2. Simplify :

3. Simplify :

Simplifying other radicals involves a similar process, and the property discussed above can be generalized for any root, which we refer to as "nth roots," where n indicates what the exponent is. For example, for a square root, n = 2, and for a cubed root, n = 3. Below are a number of properties of radicals that can be helpful for simplifying nth roots. Refer to the radicals page for more properties of radicals that can be helpful when simplifying radicals.