# Roots and Radicals

## Multiply Square Roots


### Multiply Square Roots

We have used the Product Property of Square Roots to simplify square roots by removing the perfect square factors. The Product Property of Square Roots says
We can use the Product Property of Square Roots ‘in reverse’ to multiply square roots.
Remember, we assume all variables are greater than or equal to zero.
We will rewrite the Product Property of Square Roots so we see both ways together.
So we can multiply  in this way:
Sometimes the product gives us a perfect square:
Even when the product is not a perfect square, we must look for perfect-square factors and simplify the radical whenever possible.
Multiplying radicals with coefficients is much like multiplying variables with coefficients. To multiply  we multiply the coefficients together and then the variables. The result is . Keep this in mind as you do these examples.
When we have to multiply square roots, we first find the product and then remove any perfect square factors.
The results of the previous example lead us to this property.
By realizing that squaring and taking a square root are ‘opposite’ operations, we can simplify  and get 2 right away. When we multiply the two like square roots in part (a) of the next example, it is the same as squaring.

### Use Polynomial Multiplication to Multiply Square Roots

In the next few examples, we will use the Distributive Property to multiply expressions with square roots.
We will first distribute and then simplify the square roots when possible.
When we worked with polynomials, we multiplied binomials by binomials. Remember, this gave us four products before we combined any like terms. To be sure to get all four products, we organized our work—usually by the FOIL method.
Note that some special products made our work easier when we multiplied binomials earlier. This is true when we multiply square roots, too. The special product formulas we used are shown below.
We will use the special product formulas in the next few examples. We will start with the Binomial Squares formula.
In the next two examples, we will find the product of conjugates.

### Key Concepts

1. Product Property of Square Roots If a, b are nonnegative real numbers, then

2. Special formulas for multiplying binomials and conjugates:

3. The FOIL method can be used to multiply binomials containing radicals.

### Practice Makes Perfect

Multiply Square Roots
In the following exercises, simplify.
Use Polynomial Multiplication to Multiply Square Roots
In the following exercises, simplify.
Mixed Practice
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.
ⓑ On a scale of 1–10, how would you rate your mastery of this section in light of your responses on the checklist? How can you improve this?