multiplying and dividing radicals the product and quotient properties of square roots can be used to...
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![Page 1: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/1.jpg)
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
![Page 2: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/2.jpg)
Product Rule
• Simplify radicals
• Multiply Coefficients• Multiply radicands
– “Roots” must be the same
• Simplify, if needed
4827
3933
12 9
31263
316
34
![Page 3: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/3.jpg)
Examples: Product Rule
98824924
2722
417217
34
1756372579
7573
4915715
105
![Page 4: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/4.jpg)
Quotient Rule
• Fractions made up of radicals can be simplified just like fractions
1456
1456
4 2
![Page 5: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/5.jpg)
xyyx
315 3
xyyx
315 3
25x
52 x
5x
73
97
375
yxyx
73
97
375
yxyx
2425 yx
yx25
![Page 6: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/6.jpg)
Multiply the radicals.
1. Simplify. 6 10
60 2 15
Simplify: 60
60 4 g 15
![Page 7: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/7.jpg)
6. Simplify.
Multiply the coefficients and the radicals.
2 14 3 21
42 6
6 294
6 49 g 6
6 g 7 6
![Page 8: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/8.jpg)
7. Simplify.
Divide the radicals.
108
3
108
3
36
6
![Page 9: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/9.jpg)
8. Simplify.
8 2
2 8
4
22
8 2
2 8
8
2
2
44
1
44
1
4
![Page 10: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/10.jpg)
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.
![Page 11: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/11.jpg)
Answer:
9. Simplify.
5
7
35
7
![Page 12: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/12.jpg)
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(
![Page 13: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/13.jpg)
Combining Like Terms
• Radicals & Like Terms– Same variables– Variables have the same exponents– IDENTICAL RADICALS
• Examplesxx 32&34 xy
xxyx 2
32
&252
2
![Page 14: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/14.jpg)
• Simplify radicals if possible
• Combine coefficients
3534
Radicals ARE simplified
3
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1. Simplify.
Just like when adding variables, you can only combine LIKE radicals.
5 √5
3 5 4 5 2 5
![Page 16: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/16.jpg)
Answer: 4 √7 +3 √3
2. Simplify. 6 7 3 2 7 4 3
![Page 17: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/17.jpg)
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
![Page 18: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/18.jpg)
More Radical Fun
103
210
2
1
64
63
1067
6438
64322
6462
62
SIMPLIFY
MULTIPLY
Must have Common Denominators
1010
![Page 19: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/19.jpg)
Distributive Property with Radicals
)32(2 22
64
62
)325274(32
)3(2 8 81 10 96
8 9 10 616
72 641064072
![Page 20: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/20.jpg)
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
![Page 21: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/21.jpg)
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
![Page 22: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/22.jpg)
![Page 23: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/23.jpg)
![Page 24: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/24.jpg)
Practice
Multiply: 632
Divide:6
32
Add: 1232
Subtract: 1823
Multiply: 323 323
Rationalize: )323(
3
26
15
2
34
0
5
323
![Page 25: Multiplying and Dividing Radicals The product and quotient properties of square roots can be used to multiply and divide radicals, because: and. Example](https://reader035.vdocuments.mx/reader035/viewer/2022062407/56649f4a5503460f94c6c068/html5/thumbnails/25.jpg)