multiplying and dividing radicals the product and quotient properties of square roots can be used to...
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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:
baba andb
a
b
a .
Example 1:
82 82 16 4
Example 2:
6
5
15
2
615
52
9
1)32()35(
52
3
1
Example 3:
6332 63 )32( 18 6 29 6 2 18
Product Rule
• Simplify radicals
• Multiply Coefficients• Multiply radicands
– “Roots” must be the same
• Simplify, if needed
4827
3933
12 9
31263
316
34
Examples: Product Rule
98824924
2722
417217
34
1756372579
7573
4915715
105
Quotient Rule
• Fractions made up of radicals can be simplified just like fractions
1456
1456
4 2
xyyx
315 3
xyyx
315 3
25x
52 x
5x
73
97
375
yxyx
73
97
375
yxyx
2425 yx
yx25
Multiply the radicals.
1. Simplify. 6 10
60 2 15
Simplify: 60
60 4 g 15
6. Simplify.
Multiply the coefficients and the radicals.
2 14 3 21
42 6
6 294
6 49 g 6
6 g 7 6
7. Simplify.
Divide the radicals.
108
3
108
3
36
6
8. Simplify.
8 2
2 8
4
22
8 2
2 8
8
2
2
44
1
44
1
4
Rationalizing Radicals
To simplify a fraction with a radical in the denominator, multiply the numerator and denominator by the radical.
Example 1:
2
1
2
2
2
2 Estimation is easier with rational denominators.
This process is called rationalizing the denominator.Example 2:
3
2
3
3
3
2
3
6
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.
Answer:
9. Simplify.
5
7
35
7
Radicals representing square roots of different numbers can not be gathered like this.
Adding and Subtracting Radicals
Radicals that represent the square root of the same number can be treated as a common factor.
Examples:
3 2 3 4
2 2 2 5
But simplifying sometimes results in multiples of the same radical, which can be.
Examples:
12234 34234 383)22(34
2055 5455 5255 53
Like terms can be gathered. Unlike terms can not.
3 )24(
2 3
3 6
2 )25(
Combining Like Terms
• Radicals & Like Terms– Same variables– Variables have the same exponents– IDENTICAL RADICALS
• Examplesxx 32&34 xy
xxyx 2
32
&252
2
• Simplify radicals if possible
• Combine coefficients
3534
Radicals ARE simplified
3
1. Simplify.
Just like when adding variables, you can only combine LIKE radicals.
5 √5
3 5 4 5 2 5
Answer: 4 √7 +3 √3
2. Simplify. 6 7 3 2 7 4 3
Simplify each radical.4√9•3 - 2√16 • 3 + 2√4•54 • 3√3 - 2 • 4√3+2 • 2√5
12√3 - 8√3 + 4√5Combine like radicals
4√3 + 4√5
3. Simplify. 4 27 2 48 2 20
More Radical Fun
103
210
2
1
64
63
1067
6438
64322
6462
62
SIMPLIFY
MULTIPLY
Must have Common Denominators
1010
Distributive Property with Radicals
)32(2 22
64
62
)325274(32
)3(2 8 81 10 96
8 9 10 616
72 641064072
Multiplying Binomials With Radicals
Multiplying binomials that contain radicals sometimes results in products of radicals that can be simplified.
Examples:22 )5()3( 9 - 5 4)53( )53( 1.
2)53( 2. 2)5(569 5614
3. 2)3423( 22 3 4 3 4 2 3 2 2 3
)3(16624)2(9
62466
4862418
Conjugates
Binomials of the form and dcba dcba that are identical except for the sign separating the terms are calledconjugates.
Multiplying conjugates like these together results in a rational number:
dcba dcba
Conjugates are therefore used to rationalize certain fractions.
Example:
2222 d c b a a2b - c2d
223
4
223
223
2 22 9
223 4
)2(49
2812
89
2812
2812
Practice
Multiply: 632
Divide:6
32
Add: 1232
Subtract: 1823
Multiply: 323 323
Rationalize: )323(
3
26
15
2
34
0
5
323
