Multiplying Radicals
Notes
To multiply radical expressions we will use Radical Rule 1 to obtain the following result:
x  x  x2  x
2
2
x
This will be true no matter what the radicand is.
Examples. Multiply and simplify, if possible.
1.
 5 x  4 x 
Since we are multiplying all the terms, we can rearrange them so that the coefficients and
together and the radicals are together:
 5 x  4 x   5  4  x  x
x2
 20
 20 x
2.
3x  2 x
We can apply radical rule 1 and then simplify:
3x  2 x  3x  2 x
 6x2
 6  x2
3.

2 x 35 x
x 6

We first distribute the 2 x through the parentheses. We then multiply and simplify as
above:


2 x 35 x 
2 x 3 2 x 5 x 
6 x  10 x 2 
4.
2

x 5 3 x 4
6 x  10 x

In this problem we use FOIL and then multiply.
2


x 5 3 x 4 
2 x 3 x  2 x  4  53 x  5 4
6 x 2  8 x  15 x  20 
6 x  7 x  20
In the last step, we combine the like radical terms in the middle.
Dividing Radicals
When we divide radicals, we will use the following division rule:
Division Rule:
a
a

b
b
This rule is similar to the multiplication rule for radicals, but deals with division.
We will want to express answers in simplified form. Recall that there are 3 conditions to have a
radical in simplified form:
A radical is in simplified form if:
 there are no perfect n-factors inside the radical sign
 there are no fractions inside the radical sign
 there are no radicals in the denominator of a fraction
Examples. Simplify
1.
49
25
We see that this is not in simplified form since we have a fraction inside the radical sign.
We can fix this by applying the division rule and simplifying:
49
49 7


25
25 5
2.
5
4
Again, we have a fraction inside the radical sign. We apply the division rule and simplify
the denominator. This leaves us with a radical in the numerator, which is OK, and no
radical in the denominator.
5
5
5


4
2
4
Rationalizing Denominators
Sometimes when we follow the above procedures, we are left with a radical in the denominator.
To put the radical expression in simplified form, we must eliminate the radical using a procedure
called “rationalizing the denominator”. Study the following:
Examples. Simplify.
1.
3
5
This problem has a radical in the denominator. To simplify, we begin by multiplying both
numerator and denominator by
We then simplify the result.
5 . This is the same as multiplying the fraction by 1.
3
5

5
5

3 5
25

3 5
5
Notice that our final answer does not have a radical in the denominator.
2.
5
3
We begin by using the division rule and then multiplying numerator and denominator by
3:
5
3
3.
5
3

3
3

15
9

15
3
3
18
We begin by applying the division rule:
3
3
. We notice that the radical in the

18
18
denominator has a perfect square factor, so we simplify the denominator first. We then
can rationalize the denominator by multiplying numerator and denominator by
3
18

3
9 2

3
9 2

3
3 2

2
2

6
3 4

6
6

3 2
6
2.