Download Simplifying Radicals

Warm-up
•
Simplify
1.
2.
3.
4.
62
(-14)2
-92
02
1.
2.
3.
4.
36
196
-81
0
Simplifying Radicals
Essential Question
• How do I evaluate and approximate
square roots?
Square root of a #
• If b2 = a, then b is a square root of a.
Example: if 32 = 9, then 3 is a square root of 9
Definition of Square Root
• If a is a # greater than or equal to zero,
the a represents the principal, or
positive, square root of a and a negative
sq. rt. is represented by  a
Examples:
9 =3
- 9 = -3
Radical and Radicand
• What are they?
Radical sign
 k
Positive or Negative
Radicand:
# or
expression
under
radical
symbol
Perfect Squares
• Numbers whose square roots are
integers or quotients of integers.
Examples: 4, 16, 25, 100, ¼
Example 1
• Simplify
1.
2. - 25 = -5
16 = 4
3.
1
1
=
16 4
What if the radicand is
not a perfect square?
• You can do one of 2 things…
– Give an approximation
– Give a simplified exact answer
Read the directions to see which one you should do!
Example 1
• Simplify. Give an exact answer.
8
Write the prime factorization of 8!

2 2 2
Now circle your pairs!
Pull out one number and throw out the other one.
What is left?
2 2
Example 2
• Simplify. Give an exact answer.
24
Write the prime factorization of 24!

2 2 2 3
Now circle your pairs!
Pull out one number and throw out the other one.
What is left?
2 6
Example 3
• Simplify. Give an exact answer.
80
Write the prime factorization of 80!
2 2 2 2 5
Now circle your pairs!
Pull out one number from each pair and throw out the other ones.
What is left?
4 5
Example 4
• Simplify. Give an exact answer.
5 18
 5 2 33
 5 3 2
=15 2
Product Property
• The square root of a product equals the
product of the square roots of the
factors.
a b 
ab
Example:
3  5  15
Example 1 (Simplify)
3 6
 18
Now simplify!!

33 2
3 2
Example 2 (Simplify)
a.
2 3  4 2 
8 6
b.
20

12 3 4

 60 48
 60 2  2  2  2  3
 2  2  60 3
 240 3
Distribute.
• Multiply/distribute
.

2 6  12

6 2  24
This is
simplified.
Can’t add.
Radicands are
different.
6 2  2 2 23
6 2 2 6
Use FOIL
• Multiply.
(3  2 )( 4  2 )
12  3 2  4 2  2
10  2
F irst
O utside
I nside
L ast
Always write number term before radical term!
Quotient Property
• The square root of a quotient equals the
quotient of the square roots of the
numerator and the denominator.
a

b
a
b
when a and b are positive numbers
Example:
9

25
9
3

25 5
Example 1 (Simplify)
3
49

3
49
3

7
Rationalizing
Denominators
• For example..
3
2
2
3
It is perfectly fine to
have a radical in your
NUMERATOR.
It is NOT o.k. to leave a
radical in your
DENOMINATOR!
Example 2 (Simplify)
3
2
3

2
2
2
This is just a
fancy form of
the number 1

3 2

4
3 2

2
Example 3 (Simplify)


11
3
11
3

3
3
33
9
33

3
Example 6
150
6

25
5
You don’t have to
rationalize. Just divide!!
Conjugates
Expression
Conjugate
Product
a b
a b
a b
4 x
16  x
4 x
c 2
y 7
c 2
y 7
2
c 2
2
y 7
2
Example
3
4 2
3
4 2

4 2 4 2
3( 4  2 )
4  2 (4  2 )


12  3 2
16  2
12  3 2
14
Example 5
• Give and approximation. Round to the nearest hundredth.
78
≈ 8.83
Example 6
•
Give and approximation. Round to the nearest hundredth.
18
≈ 4.24
Homework
• Page 144
– Numbers 1-12 all, 14, 16, 22.