notes 10-1 simplifying radical expressions
TRANSCRIPT
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Notes 10-1A Simplifying Radical Expressions
√
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I. Product Property of Square Roots
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Method 1: Perfect Square Method
-Break the radicand into perfect
square(s) and simplify.
72
362
36 2
6 2
163
16 3
48
4 3
A. Simplifying simple radical expressions
Ex 1: Ex 2:
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80
50
125
450
=
=
=
=
5*16
2*25
5*25
2*225
=
=
=
=
54
5 2
55
215
Perfect Square Factor * Other Factor LE
AV
E IN
RA
DIC
AL
FOR
M
Ex 3:
Ex 4:
Ex 5:
Ex 6:
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Method 2: Pair Method Sometimes it is difficult to recognize perfect squares within a number. You will get better at it with more practice, but until then, here is a second method: -Break the radicand up into prime factors -group pairs of the same number -simplify -multiply any numbers in front of the radical; multiply any numbers inside of the radical
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Example 1:
6 2
2 ∗ 2 ∗ 2 ∗ 3 ∗ 3 Step 1: Break up into prime factors
Step 2: Group together any pairs
Step 4: Multiply numbers in front of radical; multiply numbers inside radical
2 ∗ 2 ∗ 3 ∗ 3 ∗ 2
Simplify 72
2 ∗ 3 2
4 9 2 Step 3: Simplify
Ex 1:
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Example 2:
2 ∗ 2 ∗ 2 ∗ 2 ∗ 3 Step 1: Break up into prime factors
Step 2: Group together any pairs
Step 4: Multiply numbers in front of radical; multiply numbers inside radical
2 ∗ 2 ∗ 2 ∗ 2 ∗ 3
Simplify 48
2 ∗ 2 3
22 22 3 Step 3: Simplify
4 3
Ex 2:
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40
More Examples:
4 10 2 10
7 75 7 25 3 7 5 3 35 3
104
3257
Ex 3:
Ex 4:
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8
20
32
75
40
=
=
=
=
=
2*4
5*4
2*16
3*25
10*4
=
=
=
=
=
22
52
24
35
102
LEA
VE
IN R
AD
ICA
L FO
RM
Ex 5:
Ex 6:
Ex 7:
Ex 8:
Ex 9:
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B. Using Product Property to Multiply Square Roots
Ex 1: Multiply 3 ∗ 15
3 ∗ 3 ∗ 5
32 ∗ 5
3 5
Method 2: Multiply together first
3 ∗ 15
45
9 ∗ 5
3 5
3 ∗ 15
Method 1: Break down and simplify
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Practice
5 ∗ 10 2 ∗ 3 ∗ 6 ∗ 8
6 48 50
6 16 ∗ 3 2 ∗ 25
6 ∗ 4 3 5 2
Example 2: Mulitply 5 ∗ 10 Example 3: Multiply 2 6 ∗ 3 8
24 3
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II. Simplifying Radical Expressions with Variables
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More Examples:
1. 30a34
a34 30
a
1730
2. 54x4y
5z
7
9x4y
4z
6 6yz
3x
2y
2z
36yz
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A. Examples Remember!!!!!
3.
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More Examples
11x 10x x
5x x
418x 49 2x 23 2x
Ex 4:
Ex 5:
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16
81
Examples:
2
5
4
9
45
49
aIf and are real numbers and 0, then
b
aa b b
b
III. Quotient Rule for Square Roots
2
25
9 5
7
3 5
7
16
81
2
25
45
49
Ex 1: Ex 2:
Ex 3:
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15
3
90
2
3 5
3
3 5
3
5
9 10
2
9 2 5
2
9 2 5
2
3 5
Ex 4:
Ex 5:
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Examples:
1. 7
16
2. 32
25
7
16
7
4
32
25
32
5
4 2
5
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Ex 3: Simplify 48
3
48
3
16 ∗ 3
3
4 3
3
4
Ex 4: Simplify
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77
25
y
8
27
x
67
25
y y
3 7
5
y y
8
9 3
x
4
3 3
x8
27
x
Ex 4:
Ex 5:
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IV. Rationalizing the Denominator
Radical Expressions are fully simplified when: – There are no prime factors with an exponent greater
than one under any radicals
– There are no fractions under any radicals
– There are no radicals in the denominator
Rationalizing the Denominator is a way to get rid of any radicals in the denominator
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A. Denominators with one term (which is a radical)
• To rationalize the denominator of a quotient with a denominator of one term, multiply numerator and denominator by that term.
Ex 1: Ex 1: