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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

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.

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.

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.

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** 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.

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.

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.

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

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.

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

In the next example, we will find the square root of a radicand that is not a perfect square. First, we look for factors that are perfect squares and apply the Product Property. Sixteen is a factor of 160, and *y*^2 is a factor of *y*^3. The factors that are perfect squares are simplified. The remaining factors that are not perfect squares will remain in the radical.

Let's look at an example that has a fraction as its radicand.

First, we apply the Quotient Property to separate the numerator and denominator into two radicals. Next, we check to see if they can be simplified. The numerator contains factors that are perfect squares, so they can be simplified. However, the denominator does not contain any perfect squares, so it cannot be simplified.

Remember that a simplified expression cannot have a radical in the denominator. To resolve this we must multiply the numerator and denominator by the radical.

A radical expression consists of a **radical symbol**, an **index** and a **radicand**. The radicand may be a number, variable or both. The expression is considered simplified when the radicand does not have a perfect root factor and the radicand is not a fraction. If the simplified expression is a fraction, it may have a radical in the numerator, but not the denominator. Strategies for simplifying radical expressions include identifying perfect roots and applying the **Product Property** and **Quotient Property** of radicals.

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SAT Mathematics Level 2: Help and Review22 chapters | 225 lessons

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