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Algebra 2: Unit 8
Roots and Radicals
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Radicals (also called roots) are directly related to exponents.
Roots and Radicals
![Page 3: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/3.jpg)
The simplest types of radicals are square roots and cube roots.
Radicals beyond square roots and cube roots exist, but we will not discuss them as in depth.
Roots and Radicals
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The rules for radicals that you will learn work for all radicals – not just square roots and cube roots.
Roots and Radicals
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The symbol used to indicate a root is the radical symbol -
Roots and Radicals
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Every radical expression has three parts…
• Radical symbol• Index• Radicand
Roots and Radicals
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Every radical expression has three parts…
Index
Radicand
Radical
Roots and Radicals
2 49
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The index of a radical is a whole number greater than or equal to 2.
Roots and Radicals
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The index of a square root is always 2.
Roots and Radicals
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By convention, an index of 2 is not written since it is the smallest possible index.
Roots and Radicals
![Page 11: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/11.jpg)
but is normally written as .
The square root of 49 could be written as …
Roots and Radicals
2 49
49
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The index of a cube root is always 3.
All indices greater than 2 must be written.
Roots and Radicals
plural of index
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The cube root of 64 is written as
Roots and Radicals
.3 64
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What does square root mean?
What does cube root mean?
Roots and Radicals
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The square root of a number (or expression) is another number (or expression)…
Roots and Radicals
…which when multiplied by itself (squared) gives back the original number (or expression).
![Page 16: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/16.jpg)
The cube root of a number (or expression) is another number (or expression) …
Roots and Radicals
…which when multiplied by itself three times (cubed) gives back the original number (or expression).
![Page 17: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/17.jpg)
Roots and Radicals
49777 2
Example:
because
Also
because
749
749 49777 2
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Roots and Radicals
Example:
has two answers:
7 is called the positive or principal square root.
-7 is called the negative square root.
49
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Intermediate Algebra MTH04
Roots and Radicals
644444 3
Example:
because
because
4643
4643 644444 3
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What are the first 10 whole numbers that are perfect squares?
Roots and Radicals
1, 4, 9, 16, 25, 36, 49, 64, 81, 100
2222222222 10,9,8,7,6,5,4,3,2,1
![Page 21: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/21.jpg)
What are the first 10 whole numbers that are perfect cubes?
Roots and Radicals
1, 8, 27, 64, 125, 216, 343, 512, 729, 1000
3333333333 10,9,8,7,6,5,4,3,2,1
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If a number is a perfect square, then you can find its exact square root.
Roots and Radicals
A perfect square is simply a number (or expression) that can be written as the square [raised to 2nd power] of another number (or expression).
![Page 23: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/23.jpg)
Examples:principal square root
Roots and Radicals
2416 2525
22.144.1 2
11
3
121
9
416
525
11
3
121
9
2.144.1
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Examples:
Roots and Radicals
22 636 bb
236 mm
bb 636 2 36 mm
principal square root
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Roots and Radicals
If a number is a perfect cube, then you can find its exact cube root.
A perfect cube is simply a number (or expression) that can be written as the cube [raised to 3rd power] of another number (or expression).
![Page 26: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/26.jpg)
Examples:
Roots and Radicals
3464
3
5
6
125
216
32.1728.1
35125
4643
5
6
125
2163
2.1728.13
51253
principal cube root
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Examples:
Roots and Radicals
33 28 cc
3412 327 yy
326 mm
cc 283 3
43 12 327 yy
23 6 mm
principal cube root
![Page 28: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/28.jpg)
Not all numbers or expressions have an exact square root or cube root as in the previous examples.
Roots and Radicals
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If a number is NOT a perfect square, then you CANNOT find its exact square root.
You can approximate these square roots and cube roots of real numbers with a calculator.
Roots and Radicals
If a number is NOT a perfect cube, then you CANNOT find its exact cube root.
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Examples:
Roots and Radicals
40 325.6
3 74
3 40
135
198.4
42.3
619.11
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If a number is NOT a perfect square, then you might also be able to SIMPLIFY it.
What is the process to simplify a square root?
Roots and Radicals
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If the expression is not a perfect square ...
1. see if you can rewrite the expression as a product of two smaller factors...
2. where one of the factors is a perfect square.
Roots and Radicals
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3. Then, extract the the square root of the factor that is a perfect square …
4. and multiply that answer times the other factor still under the radical symbol.
Roots and Radicals
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Examples – Simplifying Square Roots: perfect square
Roots and Radicals
40 104 102
750x xx 225 6 xx 25 3
135 159 153
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If a number is NOT a perfect cube, then you might also be able to SIMPLIFY it.
What is the process to simplify a cube root?
Roots and Radicals
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If the expression is not a perfect cube ...
1. see if you can rewrite the expression as a
product of two smaller factors...
2. where one of the factors is a perfect cube.
Roots and Radicals
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3. Then, extract the the cube root of the factor that is a perfect cube…
4. and multiply that answer times the other factor still under the radical symbol.
Roots and Radicals
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perfect cubeExamples – Simplifying Cube Roots:
Roots and Radicals
3 80 3 108 3 102
3 824x 3 26 38 xx 3 22 32 xx
3 405 3 1527 3 153
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Example:
Not all square roots can be simplified!
cannot be simplified! • 77 is not a perfect square …
• and it does not have a factor that is a perfect square.
Roots and Radicals
77
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Example:
Not all cube roots can be simplified!
cannot be simplified! • 30 is not a perfect cube …
• and it does not have a factor that is a perfect cube.
Roots and Radicals
3 30
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Multiplication Division
b may not be equal to 0.
The Rules (Properties)
Roots and Radicals
baba b
a
b
a
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Multiplication Division
b may not be equal to 0.
The Rules (Properties)
Roots and Radicals
333 baba 33
3
b
a
b
a
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Multiplication Division
Examples:
Roots and Radicals
39
3333
416
6
96
6
96
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Multiplication DivisionExamples:
Roots and Radicals
3
33
3
3
333
102
108
108
80
165165
3
33
33
33
3
23
227
22754
5
270
5
270
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To add or subtract square roots or cube roots...
• simplify each radical
• add or subtract LIKE radicals by adding their coefficients.
Two radicals are LIKE if they have the same expression under the radical symbol.
Roots and Radicals
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Examples:
Roots and Radicals
6463
33 116112
67
3 114
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Example:
Roots and Radicals
3512
37
3532
3534
![Page 48: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/48.jpg)
Example:
Roots and Radicals
7135403 33
3
33
33
54
753523
7527583
![Page 49: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/49.jpg)
Conjugates
Radical conjugates are two expressions of the form .
Conjugates have the property that when you multiply them, you get a rational number – the radical is gone.
Roots and Radicals
cbacba and
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Example – Conjugates:
Roots and Radicals
735735
38
6325
7925
49971571525
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Rationalizing the Denominator
The process of removing a radical from the denominator of a fraction is called rationalizing the denominator.
Roots and Radicals
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Rationalizing the Denominator
To do this, multiply the fraction with the radical in the denominator by “1” as a fraction where the numerator and denominator are either:
• the radical factor that will produce a perfect square in the denominator radical or• the expression that is the conjugate of the denominator of the fraction to be rationalized.
Roots and Radicals
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Examples:
Roots and Radicals
6
4
3
62
6
64
36
64
6
6
6
4
3 4
102
45
4
410
4
410
44
410
4
4
4
10 3 23 2
3 3
3 2
3 23
3 2
3 2
3 2
3
![Page 54: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/54.jpg)
Example:
Roots and Radicals
102
3 102102
1023
102
102
102
3
104
1023
102
102322
2
102
6
1023
![Page 55: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/55.jpg)
Solving Radical Equations
A radical equation is simply one that has a radical term that contains a variable.
Example:
Roots and Radicals
52 c
![Page 56: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/56.jpg)
To solve a radical equation
• Get the radical term by itself on one side of the equation.
• Square both sides of the equation.
• Finish solving for the variable, if needed.
• Check your solution. This is critical when solving radical equations.
Roots and Radicals
![Page 57: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/57.jpg)
Example:
Roots and Radicals
52 c
223c
3c
9c
![Page 58: Algebra 2: Unit 8 Roots and Radicals. Radicals (also called roots) are directly related to exponents. Roots and Radicals](https://reader030.vdocuments.us/reader030/viewer/2022032414/56649efa5503460f94c0c86f/html5/thumbnails/58.jpg)
Example:
Roots and Radicals
2732 x
2532 x
532 x
22532 x
282 x
14x