Multiplying and Dividing With Square Roots
We have already looked in some detail at multiplication and
division with numerical square roots. The rules for
multiplication and division with algebraic square roots are
exactly the same as those for numerical square roots – in
fact, we stated those rules there in an algebraic form:

property 1 
and

property 2 
where ‘a’ and ‘b’ stand for any valid
mathematical expression.
When expressions involving square roots are to be multiplied
or divided, just use the above rules to combine square roots as
necessary. Then simplify the resulting expression as much as
possible using methods of simplification already described and
illustrated extensively in the preceding documents in this
series.
The examples following below illustrate this general strategy,
but also provide some examples for your own practice.
Example 1:
Simplify
solution:
Using property (1) above,
as the final answer.
Example 2:
Simplify
solution:
Using property (1) above, we get
Example 3:
Simplify
solution:
You could regard this problem as a product
and use the procedures illustrated in the previous examples.
However, an even faster way to get the answer is to distribute
the power 2 over all factors in the brackets:
as the final answer. Here we have used the basic fact thatÂ´.
Example 4:
Simplify
solution:
as the final simplified answer.
Example 5:
Simplify
solution:
This is a product of three square root factors, and so does
not fit property (1) at the beginning of this document precisely.
However, we can apply property (1) in a stepwise fashion
So
which now does match property (1). Continuing
as the final simplified result.
The obvious implication of this stepwise application of
property (1) is that property (1) can be extended to products of
any number of square root factors.
