Download Notes 10-1 Simplifying Radical Expressions

Notes 10-1A Simplifying Radical
Expressions
√
I. Product Property of Square Roots
A. Simplifying simple radical expressions
Method 1: Perfect Square Method
-Break the radicand into perfect
square(s) and simplify.
Ex 1:
72
Ex 2:
48
 36 2
 16 3
 36 2
 16 3
6 2
4 3
LEAVE IN RADICAL FORM
Perfect Square Factor * Other Factor
Ex 3:
80 =
16 * 5 =
4 5
Ex 4:
50 =
25* 2 =
5 2
Ex 5:
125=
25* 5 =
5 5
Ex 6:
450=
225* 2 = 15 2
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
Example 1:
Ex 1: Simplify
72
2∗2∗2∗3∗3
2∗2∗ 3∗3∗ 2
Step 1: Break up into prime
factors
Step 2: Group together any
pairs
4 9 2
2∗3 2
6 2
Step 3: Simplify
Step 4: Multiply numbers in
front of radical; multiply
numbers inside radical
Example 2:
Ex 2:
Simplify 48
2∗2∗2∗2∗3
2∗2∗ 2∗2∗ 3
22 2 2 3
2∗2 3
 4 3
Step 1: Break up into prime
factors
Step 2: Group together any
pairs
Step 3: Simplify
Step 4: Multiply numbers in
front of radical; multiply
numbers inside radical
More Examples:
Ex 3:
40 
4 10  4 10  2 10
Ex 4:
7 75  7 25  3  7 25 3  7  5 3  35 3
LEAVE IN RADICAL FORM
Ex 5:
8 =
4*2
=
2 2
Ex 6:
=
4*5
=
2 5
32 =
16 * 2 =
4 2
75 =
25 * 3 =
5 3
Ex 7:
Ex 8:
Ex 9:
20
40
=
4 *10 = 2 10
B. Using Product Property to Multiply Square Roots
Ex 1: Multiply 3 ∗ 15
Method 1: Break down and simplify
3 ∗ 15
3 ∗ 3∗ 5
32 ∗ 5
Method 2: Multiply together first
3 ∗ 15
45
9∗5
3 5
3 5
Practice
Example 2: Mulitply 5 ∗ 10
5 ∗ 10
Example 3: Multiply 2 6 ∗ 3 8
2∗3∗ 6∗ 8
50
6 48
2 ∗ 25
6 16 ∗ 3
5 2
6∗4 3
24 3
II. Simplifying Radical Expressions with
Variables
More Examples:
1.
30a  a  30
34
34
 a
17
2.
30
54x y z  9x y z  6 yz
4
4
5 7
4 6
 3x y z
2
2
3
6 yz
A. Examples
Remember!!!!!
3.
More Examples
Ex 4:
Ex 5:
x 
11
x x 
10
x
5
x
18x  9  2x  3x 2 2
4
4
III. Quotient Rule for Square Roots
If
a and b are real numbers and b  0, then
a
a

b
b
Examples:
Ex 1:
Ex 2:
16 4
16


81
81 9
Ex 3:
45

49
45

49
2

25
95
3 5

7
7
2
2

5
25
Ex 4:
Ex 5:
90

2
15

3
9 10

2
35
3 5


3
3
5
9 25
9 2 5

 3 5
2
2
Examples:
7
1.
7

16
7

4
16
2.
32

25
32
25

32 4 2

5
5
Ex 3: Simplify
48
3
16 ∗ 3
3
4 3
3
4
48
3
Ex 4: Simplify
Ex 4:
27

8
x
27
x
8

93
x
8
3 3
 4
x
Ex 5:
7
7y

25
7 y y
6
25

y
3
7y
5
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
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: