Intermediate Algebra
Section 7.4 – Adding, Subtracting, and Multiplying
Radical Expressions
We have seen like polynomial terms and learned how to combine
them using the distributive property. For example,
7x + 3x = (7 + 3)x + 10x . We use the same property for radical
expressions, so 7 5 + 3 5 = (7 + 3) 5 = 10 5 . The radical
expressions 7 5 and 3 5 are called like radicals.
Like Radicals
Radicals with the same index and the same radicand are like radicals.
Only like radicals can be added or subtracted. For instance, we
cannot combine 4 2 and 4 3 since they are not like terms because
the radicands are not the same, and we cannot combine 7 and 3 7
since the indices are not the same.
If necessary, we should simplify the radicands and then determine if
we can combine like terms.
Example:
a)
Add or subtract.
2 6+8 6−3 6
b)
5 12 + 16 27
Section 7.4 – Adding, Subtracting, and Multiplying Radical Expressions
c)
3 7 − 3 x + 4 7 − 33 x
d)
page 2
x 3 16 x5 + 2 3 54 x8
We can multiply radical expressions using many of the same
properties that we used to multiply polynomial expressions. These
include the distributive property and FOIL.
Example:
a)
b)
5y
Multiply the following.
(
y+ 5
)
(1 − 3 )( 2 + 3 )
Section 7.4 – Adding, Subtracting, and Multiplying Radical Expressions
c)
(4
)(
d)
(4 − 2 5 )
e)
(
x +3 3 x +2
10 + 7
page 3
)
2
)(
10 − 7
)
Recall that the product of the sum and difference of two terms is
( a + b )( a − b ) = a 2 − b 2 . When dealing with radical expressions, the
terms
a + b and
a − b are called conjugates (Also,
a +b
and a − b are also conjugates, so you don’t need a radical in both
terms to be conjugates). We will need to know about conjugates
when we are dividing rational expressions.