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Unit 4: Radicals and Roots
Lesson1: Perfect Squares
Squared: to raise a number to the second power or have an exponent of 2
Example: 4 raised to the second power is 4 squared or 42
42 = 4 x 4 = 16
122 = 12 x 12 = 144
Perfect Square: Any number whose square root is a whole number
Example:
36 is a perfect square because √36 = 6 (whole number)
12 is not a perfect square because √12 = 3.46
(not a whole number)
First 12 perfect squares:
12 = 1
22 = 4
32 = 9
42 = 16
52 = 25
62 = 36
72 = 49
82 = 64
92 = 81
Examples and Notes from Class:
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102 = 100
112 = 121
122 = 144
Lesson 2: Perfect Cubes
Cubed: to raise a number to the third power or have an exponent of 3
Example: 6 raised to the third power is 6 cubed or 63
63 = 6 x 6 x 6 = 256
33 = 3 x 3 x 3 = 27
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To Write a Cube Root: √125
itself equals 125?
this means, what number, multiplied 3 times by
3
5 x 5 x 5 = 125. So, √125 = 5
Perfect Cube: Any number whose cube root is a whole number
Example:
3
64 is a perfect cube because √64 = 4 (4 x 4 x 4 = 64)
3
100 is not a perfect cube because √100 = 4.64
(not a whole number)
First 6 perfect Cubes:
13 = 1 x 1 x 1 = 1
23 = 2 x 2 x 2 = 8
33 = 3 x 3 x 3 = 27
43 = 4 x 4 x 4 = 64
53 = 5 x 5 x 5 = 125
63 = 6 x 6 x 6 = 256
Notes from Class:
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Lesson 3: Simplify Roots (Radicals)
Radical: another name for the square root symbol
Evaluate Square Roots
√64 = 8 because 8 x 8 = 64
√64 = -8 because -8 x -8 = 64
So, we write: √64 = ±8 (this is read, the square root of 8 equals plus or minus 8)
Principal Root: the positive solution to a square root problem
The principal root of 63 is +8
Evaluate Cube Roots
3
√256 = 6 because 6 x 6 x 6 = 256
3
√256 ≠ (does not equal) - 6 because -6 x -6 x -6 = -256
Example Evaluate and Order Roots:
Evaluate:
Order from Smallest to Largest:
4
3
3
√81, √64 , √4 , √256
Notes and Examples from Class
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Lesson 4: Identify Irrational Numbers
Rational Number: can be expressed as the ratio of 2 integers (fraction)
As a decimal, a rational number terminates OR repeats
Examples:
8 is rational because it can be expressed as
8
1
-34 is rational because it can be expressed as −
.35 is rational because it can be expressed as
34
1
35
100
AND because it terminates
.124124124… is rational because it is a repeating decimal
Irrational Number: cannot be expressed as the ratio of 2 numbers (fraction)
As a decimal it does NOT terminate OR Repeat
Examples: .543823…
-63.847362514…
π (= 3.141592…)
Example:
√400 = ±20
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Example:
√48 = 6.928203…
Expressions: Rational or Irrational?
Add, Subtract, Multiply or Divide 2 Rational Numbers: answer is Rational
Example: √36 ÷ 5
√36 is rational and 5 is rational so the answer is rational
Add, Subtract, Multiply or Divide an Irrational Number and a Rational Number:
answer is Irrational
Example: √40 ÷ 5
√40 is irrational and 5 is rational so the answer is irrational
Notes and Examples from Class
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Lesson 6: Simplify Radicals
Prime Numbers: any number whose ONLY factors are 1 and itself
Example: 2 (because the only numbers you can multiply to get 2 are 2 and 1)
13 (because the only numbers you can multiply to get 13 are 13 and 1)
Prime Factorization: break down a number into its prime factors
40 = 2 x 2 x 2 x 5
Create a Factor Tree to Prime Factor the number 40
Simplify Square Roots:
Example 1: Simplify √24
Prime Factor 24
= 2 √6
Take out 1 of any pair
Example 2:
Prime Factor 180
= 6√5 Take out 1 of any pair
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Simplify Cube Roots
3
Example: √120
Prime Factor 120 using a Factor Tree:
3
√120 =
Rewrite the problem
Take out 1 of any triplet of the same number
3
3
√120 = 2 √15
10
Notes and Examples from Class:
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Lesson 7: Multiply and Divide Radical in an Expression
Coefficient: the number in front of the radical
Radicand: the number inside the radical
Example: 5√7
5 is the coefficient and 7 is the radicand
Multiply Radicals: Multiply the coefficients and the radicands separately, then
simplify
Example:
4x2=8
Multiply the coefficients
√8 x √3 = √24
Multiply the radicands
Simplify:
Examples from Class:
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Dividing Radicals: Divide the coefficients and radicands separately, then simplify
Example:
Divide 32 by 8 and 24 by 2
Simplify √12
Simplify
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Multiplying Radicals and the Distributive Property
Notes and Examples from Class:
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Lessons 8: Add and Subtract Radicals
Radicands (the number under the radical sign) MUST be the same in order to add
or subtract
Examples:
If the radicands aren’t the same, simplify to see if you can make them the same
Example: 5√18 + 2√50
Simplify
Add
Combine Like Terms to Add or Subtract: (Terms with the same Radicand)
Notes and Examples from Class:
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Lessons 9: Rationalize the Denominator of an Expression
Simplified Fraction: cannot have a radical in the denominator
Examples of Removing the Radical in the Denominator
Example 1:
Find the square root of the denominator
Example 2:
Multiply the top and bottom times the square root
√6 ÷ √6 = 1, so this will not change the value of the fraction
The final, simplified answer may have a radical in the numerator, but not the
denominator
Example 3:
The 10 on the top and bottom cancel each other out
Notes and Examples from Class
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