When multiplying and dividing radical expressions, we use many of the same properties we learned

previously. The main property we will use are the distributive property as well as the FOIL method. After applying these, simplify the radicals and combine like terms. The first important rule when multiplying radical expressions is that the index of the terms being multiplied must be the same. For example, we can multiply √2 and √10 but we cannot multiply √2 and 3√10.

**Example:** Multiply and simplify.

4√10 ∙ 5√5

4 ∙ 5 = 20 , √10 ∙ √5 = √50 = √25 ∙ √2 = 5√2

Therefore 4√10 ∙ 5√5 = 20 ∙ 5√2 = 100√2

**Example**: Multiply and simplify.

√(20×3) ∙ √(5xy2)

√20 ∙ √5 = √100, √(x3) ∙ √x ∙ √(y2) = √(x4) ∙ √(y2) = 10 = x2y

Therefore, √(20×3) ∙ √(5xy2) = 10x2y.

**Example:** Multiply and simplify.

3√(5xy4) ∙ 3√(25x2y5)

3√5 ∙ 3√25 = 3√125 = 5,

3√(x) ∙ 3√(x2) = 3√(x3) = x,

3√(y4) ∙ 3√(y5) = 3√(y9) y3

Therefore, 3√(5xy4) ∙ 3√(25x2y5) = 5xy3.

We’ll now show an example where you use the distributive property to multiply a radical expression with one term by a radical expression with two or more terms.

**Example:** Simplify 4√5(3√7 + 2√2)

Distribute the 4√5 to get 4√5(3√7) + 4√5(2√2).

12√35 + 8√10 (multiply the coefficients and multiply the radicals)

If we want to multiply two radical expressions with two terms each, we use the FOIL method. Then we

simplify and combine like terms.

**Example:** Multiply (√5 – √2)(√5 + 3√6)

Recall that the FOIL method is multiplying the First terms, then the Outer terms, the Inner terms and then the Last terms. Another way to think of it is to multiply each term in the first set of parentheses by each term in the second set of parentheses.

(√5 ∙ √5) + (√5 ∙ 3√6) + ( – √2 ∙ √5) + ( – √2 ∙ 3√6)

5 + 3√30 – √10 – 3√12

5 + 3√30 – √10 – 3(√4 ∙ √3)

5 + 3√30 – √10 – 3(2 · √3)

5 + 3√30 – √10 – 6√3

We now focus on the division of radicals. The important aspect with division of radicals is to make sure there are no radicals in the denominator of the fraction. We remove radicals from the denominator by what is known as rationalizing the denominator. What this means is that we replace the radical in the denominator by a rational number. For example, if you want to divide √6 by √2, first write this as a fraction to eliminate √2 from the denominator, we need to multiply the numerator and the denominator by something that will make the number under the radical a perfect square. In this case we know that 4 is a perfect square, so multiply the numerator and denominator by √2 to get

(√ 6/√2) ∙ (√2/√2 ) = (√12/√4) = (√4∙√3)/√ 4 = √3.

Note that when you need to rationalize the denominator that has a square root, multiply the numerator and denominator by the denominator. The result of the denominator is the term under the radical. For example, if the denominator is √(6x), multiply the numerator and denominator by √(6x) and the denominator becomes 6x. Knowing this fact will make the process easier by cutting out the step of multiplying √(6x) by √(6x) to get √(36x)2 and then simplifying to get 6x.

**Examples**: Rationalize each denominator.

1. √(10/3)

Begin by writing this as a fraction of two radicals and multiply the numerator and denominator by √3.

(√10/√3) = (√10/√3) ∙ (√3/√3) = √30/√9 = √30/3

2. 6√3 / (5√2)

Begin by writing this as a fraction of two radicals and then multiply the numerator and denominator by √2.

6√3 / 5√2 ∙ (√2/√2) = (6√6)/(5(2)) = (6√6)/10 = (3√6)/5

3. √(3 xy2) / √(2x)

Multiply the numerator and denominator by √(2x) because 2x ∙ 2x = 4×2 which is a perfect square.

√(3 xy2) / √(2x) ∙(√(2x)) / √(2x) = √(6 x2 y2 ) / √(4×2) = (xy √6) / (2x) = (y√ 6)/2

Suppose the denominator is a cube root. To rationalize a cube root, we need to multiply the

denominator by something that will make what’s under the radical a perfect cube. For example, suppose the denominator of a radical expression is 3√9. To rationalize this, we must multiply the numerator and denominator by 3√3 because 3√9 ∙ 3√3 = 3√27 = 3.

This guide should help clear any confusion involving multiplying and dividing radical expressions.