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Multiplying Radical Expressions

Multiplying Radical Expressions

Author: Sophia Tutorial
Description:

Multiply binomial radical expressions using FOIL.

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Tutorial
what's covered
  1. Review of Distributive Rule and FOIL with Integers
  2. Multiplying Radical Expressions using Distribution and FOIL

1. Review of Distributive Rule and FOIL with Integers

The distributive rule allows us to distribute an outside factor into all terms of another factor. For example:

2 open parentheses 4 plus 3 close parentheses
Distribute 2 into 4 and 3
open parentheses 2 times 4 close parentheses plus open parentheses 2 times 3 close parentheses
Multiply inside the parentheses
8 plus 6
Add 8 to 6
14
Our Solution

If we have to factors in the form (a+b), we can use the distributive property in a different way, commonly referred to as the FOIL method.

open parentheses 6 minus 3 close parentheses open parentheses 3 plus 2 close parentheses
Apply steps to FOIL
18 plus 12 minus 9 minus 6
Add and subtract
15
Our Solution


2. Multiplying Radical Expressions using Distribution and FOIL

The distributive rule and FOIL method can be applied to multiply expressions with radicals as well. First, we will look at an example of distribution, where two identical radicals are multiplied together.

square root of 2 open parentheses 3 plus square root of 2 close parentheses
Distribute the square root of 2
3 square root of 2 plus square root of 2 square root of 2
square root of 2 square root of 2 simplifies to the integer 2
3 square root of 2 plus 2
Our Solution

We can also use the FOIL method to distribute across two binomials in multiplication when there are radicals. This is illustrated in the following example:

open parentheses square root of 3 plus 5 close parentheses open parentheses 2 minus square root of 3 close parentheses
Apply the steps to FOIL
2 square root of 3 minus square root of 3 square root of 3 plus 10 minus 5 square root of 3
square root of 3 square root of 3 simplifies to the integer 3
2 square root of 3 minus 3 plus 10 minus 5 square root of 3
Combine like terms
7 minus 3 square root of 3
Our Solution

big idea
The distributive rule and the FOIL method can be applied to expressions containing radicals as well. When two identical square roots are multiplied by each other, it evaluates to the expression underneath the square root. This property also applies to other roots, such as cube roots, but the identical radical needs to be multiplied by itself 3 times, and so on.

term to know
FOIL
An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last.

summary
A review of the distributive property and FOIL allows us to distribute outside factors into all terms of another factor. A helpful hint when multiplying radical expressions using distribution and FOIL is to remember that multiplying 2 square root terms together will cancel out the square root operation. For example, the square root of 8 times the square root of 8 is just 8.

Terms to Know
FOIL

An acronym to remember the steps for distributing factors in binomial multiplication: first, outside, inside, last. 

Formulas to Know
FOIL Method

open parentheses a plus b close parentheses open parentheses c plus d close parentheses equals a c plus a d plus b c plus b d