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.
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