Simplifying Radical Expressions with Variables

Instructor: Mia Primas

Mia has taught math and science and has a Master's Degree in Secondary Teaching.

What do you think of when you hear the word 'radical'? You may think of something that is extreme or even a person that is a political activist. In this lesson, you will learn what a radical is in math and how to simplify radical expressions.

What is a Radical Expression?

There are three parts of a radical expression that are important to understand. The radical symbol, which means ''root of,'' distinguishes radical expressions from other types of expressions. The number under the radical symbol is called the radicand, which is the number or expression you are finding the 'root of,' while the smaller number written with the radical symbol is called the index. If there is no number written as the index, it is implied that the index is two. In this case, you would take the square root of the radicand. If the index is three, you would take the cube root of the radicand.


Parts of a Radical
Parts of a Radical

Simplifying Radical Expressions

A radical expression is considered simplified when there are no perfect root factors left in the radical. For example, 36 should not be left in a square root radical because 36 is a perfect square and would be simplified to six. In the same manner, the square root of x^2 would be simplified to x, because x^2 is a perfect square.


rad1a


Examples of perfect squares
rad1b

It is also important to make sure that there are no fractions left in a radical and that fractions do not have radicals in their denominator.

When simplifying radical expressions, there are two properties that are especially useful: The Product Property of Radicals and the Quotient Property of Radicals.

The Product Property of Radicals

The Product Property of Radicals comes in handy when the radicand is not a perfect root, but has a factor that is a perfect root. In this case we can take the square root of each factor.


Product Property of Radicals
rad2

To simplify the square root of 75, we can factor 75 into 25 x 3, then take the square root of each factor. The square root of 25 is five and three remains in the radical because it is not a perfect square.


Example of the Product Property of Radicals
rad3

The Quotient Property of Radicals

The Quotient Property of Radicals is useful for radicands that are fractions. The numerator and denominator can be separated into their own radicals that can be simplified.


Quotient Property of Radicals
rad4

To simplify the square root of 25/9, we can take the square root of 25 and the square root of 9.


Example of the Quotient Property of Radicals
radquot

The simplified answer can remain as a fraction as long as the denominator does not have a radical. If it does contain a radical, the numerator and denominator can both be multiplied by the radical that is in the denominator.

In the following example, the denominator has a square root of 15. The numerator and denominator of the fraction are multiplied by the square root of 15. Notice that the radical in the denominator is eliminated because the square root of 15 times the square root of 15 is just 15. Even though there is a radical in the numerator, the expression is still considered to be simplified.


Simplifying an expression with a radical in the denominator
rad5

Radicands with Variables

The radicand may be a number, a variable or both. No matter what the radicand is, the radical symbol applies to every part of the radicand.

To simplify the square root of 36x^2, we can take the square root of the factors, which are 36 and x^2.


Example with a radicand that is a perfect square
rad6

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