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Multiplying and Dividing Radicals
The product and quotient properties of square roots can be used to
multiply and divide radicals, because:
a  b  a b
and
a
a

b
b
.
Example 1:
2 8
2 8
16
4
Example 2:
2
5

15 6
25
15  6
25
(5  3)  (2  3)
1
9
1
3
Example 3:
2 3 3 6
(2  3) 3  6
6 18
6 92
18 2
Product
Rule
• Simplify radicals
• Multiply Coefficients
• Multiply radicands
– “Roots” must be the
same
• Simplify, if needed
27  48
9 3  16 3
3 3
4 3
12 9
12 3
36
Examples: Product Rule
8  98
4  2  49  2
2 2 7 2
17 4
17  2
34
63  175
9  7  25  7
3 7 5 7
15 49
15 7
105
Quotient Rule
• Fractions made up of radicals can be simplified
just like fractions
56
14
56
14
4
2
3
15 x y
3 xy
15 x 3 y
3 xy
5x
2
x 5
75 x 7 y 9
3x 3 y 7
75 x 7 y 9
3 7
3x y
2
25 x 4 y 2
x 5
2
5x y
6  10
1. Simplify.
Multiply the radicals.
Simplify:
60
60  4 g15
60  2 15
6. Simplify.
2 14  3 21
Multiply the coefficients and the
radicals.
6 294
6 49 g 6
6 g7 6
42 6
7. Simplify.
108
3
Divide the radicals.
108
3
36
6
8. Simplify.
8 2
2 8
8 2
8 2
1
1

4
4
4
2 8 2 4
4
4
2
2
Rationalizing Radicals
To simplify a fraction with a radical in the denominator, multiply
the numerator and denominator by the radical.
Example 1:
1
2

2
2
2
2
Estimation is easier with
rational denominators.
This process is called rationalizing the denominator.
Example 2:
2
3
2 3

3 3
6
3
Since the square root of a quotient is a quotient of square roots, the
square root of a fraction must be rationalized to be in simplest form.
5
7
9. Simplify.
Answer:
35
7
Adding and Subtracting Radicals
Radicals that represent the square root of the same number can be
treated as a common factor.
Examples:
4 3  2 3
( 4  2) 3
6 3
(5  2) 2
3 2
5 2 2 2
Radicals representing square roots of different numbers can not
be gathered like this. But simplifying sometimes results in
multiples of the same radical, which can be.
Examples:
4 3  2 12
4 3  2 43
4 3  (2  2) 3
5 5  20
5 5  45
5 5 2 5
Like terms can be gathered. Unlike terms can not.
8 3
3 5
Combining Like Terms
• Radicals & Like Terms
– Same variables
– Variables have the same exponents
– IDENTICAL RADICALS
• Examples
4 3x &  2 3x
5x2
 2x2
2 xy &
2 xy
3
• Simplify radicals if
possible
• Combine coefficients
4 35 3
Radicals ARE simplified
 3
1. Simplify.
3 54 52 5
Just like when adding variables,
you can only combine LIKE
radicals.
5 √5
2. Simplify.
6 7  32 7 4 3
Answer: 4 √7 +3 √3
3. Simplify.
4 27  2 48  2 20
Simplify each radical.
4√9•3 - 2√16 • 3 + 2√4•5
4 • 3√3 - 2 • 4√3+2 • 2√5
12√3 - 8√3 + 4√5
Combine like radicals
4√3 + 4√5
More Radical Fun
1
2
10 
10
2
3
Must have
Common
Denominators
3
4

10
10
6
6
7
10
6
SIMPLIFY
8 34 6
2 2 34 6
MULTIPLY
2 64 6
2 6
Distributive Property with Radicals
2 ( 2  3)
2 2
 2 ( 3)
4 6
2 6
2 3(4 27  5 32 )
8 81 10 96
8  9  10 16 6
72  10 4 6
72  40 6
Multiplying Binomials With Radicals
Multiplying binomials that contain radicals sometimes results in
products of radicals that can be simplified.
Examples:
1. (3  5 ) (3  5 )
2
2. (3  5 )
2
3. (3 2  4 3 )
(3) 2  ( 5 ) 2
9  6 5  ( 5 )2
9-5
4
14  6 5
 3 2   2 3 2  4 3   4 3 
2
9(2)  24 6  16(3)
18  24 6  48
66  24 6
2
Conjugates




Binomials of the form a b  c d
and a b  c d
that are identical except for the sign separating the terms are called
conjugates.
Multiplying conjugates like these together results in a rational
number:
a
b c d
 a
b c d

a
2
 b  c  d 
2
2
2
a2 b - c2 d
Conjugates are therefore used to rationalize certain fractions.
Example:



4
3 2 2

3 2 2
3 2 2


12  8 2
98


4 3 2 2
9 2 2 2


12  8 2
12  8 2
9  4(2)
Practice
Multiply:
2 3 6
6 2
2 3
6
Divide:
2
Add:
2 3  12
4 3
Subtract:
3 2  18
0
Multiply:
Rationalize:
3
2 3
 3
3
(3 2  3 )
2 3

15
3 2 3
5