Multiplying Radical Expressions with Two or More Terms

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  • 0:02 Multiplying Radical…
  • 0:34 The Distributive Property
  • 2:48 Multiplying Two Binomials
  • 6:04 Lesson Summary
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Instructor: Jennifer Beddoe

Jennifer has an MS in Chemistry and a BS in Biological Sciences.

Multiplying radical expressions with more than two terms can be confusing. This lesson will take some of the confusion away by giving clear steps for multiplying these expressions. It will also provide some examples to help solidify the steps.

Multiplying Radical Expressions

Multiplying radical expressions is not much different from multiplying any other expression except the terms are under a square root symbol, which adds a few extra steps. To multiply two or more of the same variable, you need to add the exponents of the variables.

For example, 5bc^2 * 3(b^3)c = 15(b^4)(c^3).

Variables that are different, like b and c, cannot be combined.

The Distributive Property

When multiplying a single-term radical expression by a multiple-term expression, you need to use the distributive property. The distributive property means that you multiply the single term to both of the terms in the multiple-term expression. It might be easier to see with an example.

Simplify sqrt 2x(sqrt 3 + y).

First, multiply the number outside the parenthesis to the first term inside the parentheses. sqrt 2x * sqrt 3 = sqrt 6x.

Then, multiply the number outside the parenthesis to the second term inside the parentheses. sqrt 2x * y = ysqrt 2x.

Then, write out both terms. Remember, they cannot be combined unless their variables are exactly the same. (sqrt 6x) + y sqrt 2x.

Let's try another one. Simplify 2 sqrt x * (sqrt 7xy - sqrt y).

Multiply the outside term by the first inside term. 2 sqrt x * sqrt 7xy = 2 sqrt 7(x^2)y.

Then, multiply the outside term by the second inside term. 2 sqrt x * sqrt y = 2 sqrt xy.

Next, write down the two terms, making sure to keep the subtraction sign in the middle. 2 sqrt 7(x^2)y - 2 sqrt xy.

Before we can finish this one, we notice that there is an x^2 inside the radical of the first term. That can be simplified further because the square root of x^2 is x, so we can move an x outside the radical, and this problem is complete.

2x sqrt 7y - 2 sqrt xy.

Multiplying Two Binomials

When you are multiplying two binomials, expressions with two terms, you also use the distributive property, but it becomes more complex because you have to make sure that each term in the first expression is multiplied by every term in the second expression.

Mathematicians have come up with a mnemonic to help us remember what terms to multiply together. The mnemonic is FOIL, and it stands for:

  • F = the first term in each binomial
  • O = the outside term in each binomial
  • I = the inside term in each binomial
  • L = the last term in each binomial

Here is an example using the FOIL method of multiplying.

(2 + sqrt x)(5 - sqrt 3).

Start by multiplying the first term in each set of parentheses. 2 * 5 = 10.

Then, the outside term in each set of parentheses: 2 * -sqrt 3 = -2 sqrt 3.

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