College Prep 8.4 and 8.5 Notes
Adding, Subtracting, Multiplying and Dividing Radical Expressions
Adding and Subtracting Radical Expressions
Like Radicals: Radicals with the same index and the same radicand are called like radicals.
eg) 3 and 2 3; 5 7 x and  3 5 7 x
To add or subtract radicals, completely simplify the radicals, then combine like terms.
Examples: Simplify by combining like radical terms, if possible.
a) 4 11  8 11
b) 5 3 3x  7 3 3x
c) 9 4 3x  7 x 4 3x  4 4 3x
d) 4 3 10t  5 3 10t  7 10t
f)
g)
j)
3
e)
20  2 45
27 x  2 9 x  72 x
h) 9 5  4 3
p 4 q 7 - 3 64 pq
k) 4
3
54  5 3 16  3 2
i) 5 3 16x7 + 2 3 128x 4
32
18
-5
36
72
l)
60
64
+ 12
4
y
y
Products and Quotients of Two or More Radical Terms: Radical expressions often contain factors that have
more than one term. Multiplying such expressions is similar to finding products of polynomials.
Multiplying radicals: Use the same methods used to multiply polynomial expressions (distribution and FOIL).
Examples: Multiply.
a) 5 x  10

(


)(
c) 5 3 4 - 3 3 9 4 3 3 + 2

b) 3 m  4 n 7 m  2 n
)
d)

x 2 3

2



e) 3  5 3  5

f)

x 2 3

x 2 3

Rationalizing Denominators or Numerators with Two Terms:
To rationalize a numerator or denominator that is a sum or difference of two terms we use conjugates.
Conjugates are pairs of expressions with the same terms, but separated by opposite operations ( and ) .
e.g.) a  b and a  b; 2  7 and 2  7; w x  y z and w x  y z .
Examples: Rationalize each denominator.
32
a)
27
b) 3
128
375x 2
3
x 5
c)
7
2 3
d)
e)
3 7
8 5
f)
2 6 3 2
5 2 2 3
b)
3  15 2
6
Putting in Lowest Terms
a)
8  12 3
16
HW Problems: section 8.4: 15,18,21,24,27,30,33,36,45,48,51 AND
section 8.5 13,17,23,27,39,43,49,51,55,63,71,81,83,89,93,103,105