# Roots and Radicals

## Simplify Square Roots

In the last section, we estimated the square root of a number between two consecutive whole numbers. We can say that  is between 7 and 8. This is fairly easy to do when the numbers are small enough that we can use .
But what if we want to estimate ? If we simplify the square root first, we’ll be able to estimate it easily. There are other reasons, too, to simplify square roots as you’ll see later in this chapter.
A square root is considered simplified if its radicand contains no perfect square factors.
So  is simplified. But  is not simplified, because 16 is a perfect square factor of 32.

### Use the Product Property to Simplify Square Roots

The properties we will use to simplify expressions with square roots are similar to the properties of exponents. We know that . The corresponding property of square roots says that .
We use the Product Property of Square Roots to remove all perfect square factors from a radical. We will show how to do this in .
Notice in the previous example that the simplified form of  is , which is the product of an integer and a square root. We always write the integer in front of the square root.
We could use the simplified form  to estimate . We know 5 is between 2 and 3, and  is . So  is between 20 and 30.
The next example is much like the previous examples, but with variables.
We follow the same procedure when there is a coefficient in the radical, too.
In the next example both the constant and the variable have perfect square factors.
We have seen how to use the Order of Operations to simplify some expressions with radicals. To simplify  we must simplify each square root separately first, then add to get the sum of 17.
The expression  cannot be simplified—to begin we’d need to simplify each square root, but neither 17 nor 7 contains a perfect square factor.
In the next example, we have the sum of an integer and a square root. We simplify the square root but cannot add the resulting expression to the integer.
The next example includes a fraction with a radical in the numerator. Remember that in order to simplify a fraction you need a common factor in the numerator and denominator.

### Use the Quotient Property to Simplify Square Roots

Whenever you have to simplify a square root, the first step you should take is to determine whether the radicand is a perfect square. A perfect square fraction is a fraction in which both the numerator and the denominator are perfect squares.
If the numerator and denominator have any common factors, remove them. You may find a perfect square fraction!
In the last example, our first step was to simplify the fraction under the radical by removing common factors. In the next example we will use the Quotient Property to simplify under the radical. We divide the like bases by subtracting their exponents, .
Remember the Quotient to a Power Property? It said we could raise a fraction to a power by raising the numerator and denominator to the power separately.
We can use a similar property to simplify a square root of a fraction. After removing all common factors from the numerator and denominator, if the fraction is not a perfect square we simplify the numerator and denominator separately.
Be sure to simplify the fraction in the radicand first, if possible.

### Key Concepts

1. Simplified Square Root  is considered simplified if  has no perfect-square factors.
2. Product Property of Square Roots If a, b are non-negative real numbers, then

3. Simplify a Square Root Using the Product Property To simplify a square root using the Product Property:


4. Quotient Property of Square Roots If a, b are non-negative real numbers and , then

5. Simplify a Square Root Using the Quotient Property To simplify a square root using the Quotient Property:



### Practice Makes Perfect

Use the Product Property to Simplify Square Roots
In the following exercises, simplify.
Use the Quotient Property to Simplify Square Roots
In the following exercises, simplify.

### Everyday Math


### Writing Exercises


### Self Check

ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section.
ⓑ After reviewing this checklist, what will you do to become confident for all objectives?