Section 10.5
Expressions Containing
Several Radical Terms
Definition
Like Radicals are radicals that have the same index
and same radicand.
 We can ONLY combine Like Radicals.
To add/subtract radical expressions, we
•
•
1) Simplify each radical.
2) Combine like radicals.
Example
Simplify by combining like radical terms.
a) 3 5  7 5
3
2
3
2
3
b) 7 9s  9s  2 9s 2
Solution
a) 3 5  7 5  (3  7) 5= 10 5
3
2
3
2
3
3
2
b) 7 9s  9s  2 9s  (7  1  2) 9s
3
 6 9s 2
2
Example
Simplify by combining like radical terms.
a) 2 18  7 2
3
4
3
b) 10m 3m  24m
Solution
a) 2 18  7 2  2 9  2  7 2
 2(3) 2  7 2  6 2  7 2   2
3
3
b) 10m 3m  24m4  10m 3 3m  2m 3 3m
 12m 3 3m
Examples
Simplify the following expressions
5 a 3 a 7 a
3 75  2 12  2 48
4 x  2x x
3
4
3
32x  50x  18x
3
3
4 x 5 x  2
3
3
Product of two or more radical terms
1. Use distributive law or FOIL
n
2. Use product rule for radicals
3. Simplify and combine like terms.
ab  n a  n b
n
x x
n
Examples: Multiply. Simplify if possible. Assume all
variables are positive
2( y  2)
a)
b)

c)


3 x  2  3 x2  3 


m n


m n

Solution
Using the
distributive law
2( y  2)  2  y  2  2
a)
 y 2 4  y 22
b)


F
O
I
L
3 x  2  3 x2  3   3 x 3 x2  33 x  23 x 2  6




3 3
3 2
3
x 3 x  2 x 6
3 2
3
 x3 x  2 x 6
Solution
c)

m n

m n
F

O
2
I
L
 m  m n m n n
2
 mn
Notice that the two middle terms are opposites, and the
result contains no radical. Pairs of radical terms like,
m  n and m  n ,
are called conjugate pairs.
Rationalizing Denominators
with Two Terms
 The sum and difference of the same terms are
called conjugate pairs.
 To rationalize denominators with two terms,
we multiply the numerator and denominator by
the conjugate of the denominator.
Example
Rationalize the denominator:
3
5 2
Solution
3
3


5 2
5 2
5 2
5 2
3( 5  2)

25  10  10  4
1
3( 5  2) 3( 5  2)


 5 2
52
3
1
Example
5
.
7y
Rationalize the denominator:
Solution
5
5
7y

.
7y
7y 7y

5



7y
7y

5 7  5y
7 y
2

7y


5( 7  y)
49  y 7  y 7  y 2
Example
Rationalize the denominator:
4 m
m n
Solution
m n
m n
4 m
4 m

.
m n
m n

2
4 m  4 mn
2
m  mn  mn  n
4m  4 mn

mn
2
Terms with Differing Indices
To multiply or divide radical terms with different indices,
we can convert to exponential notation, use the rules for
exponents, and then convert back to radical notation.
Example
Multiply and, if possible, simplify:
5 3
x x .
Solution
5 3
1/ 2
x x x
3/ 5
x
11/10
x

10 11
x
10 10 10
 x  x
 x10 x
Converting to exponential notation
Adding exponents
Converting to radical notation
Simplifying
Group Exercise
Simplify the following radical expressions
3
2y

3
3y  3 4 y
2
( 3x  2)
6
7 4
5a
4
2a
2
2
