# Multiplying and Dividing With Square Roots

Remember the rules for multiplication and division with algebraic square roots:

**Property 1:**

**Property 2:**

where ‘a’ and ‘b’ stand for any valid mathematical expression.

## Examples with solutions

**Example 1: **

Simplify

**solution: **

as the final answer.

**Example 2: **

Simplify

**solution: **

Divisions can be written as fractions, so the methods of the last two examples can be used here.

as the final answer.

**Example 3: **

Simplify

**solution: **

This is really more of a “rationalize the denominator” problem than it is a dividing problem. We get

as the final, simplest result.

**Example 4: **

Simplify

**solution: **

as the final simplified result with the denominator rationalized.

**Example 5: **

Expand and simplify the product:

**solution: **

The final result here may be a bit of a surprise. Basically, we are asked to multiply one trinomial by another, and simplify the result. The multiplication step is a bit tedious, but you are well familiar with the method:

as the final simplified answer. Notice that all of the square root terms in the second last line have cancelled out, because they each occur in pairs of opposite sign.

If you attempt to factor the trinomial x^{ 2} + 8x +
10 into a product of two binomials there, you will fail. (Why?)

This example here shows that x^{ 2} + 8x + 10 can be
“factored” into a product, but it is a product of two
trinomials (hardly a simplification!) and those trinomials
involve the square root of a number. You can see that the
systematic trial and verification method we used to factor
trinomials into products of binomials will not work to get this
factorization.