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Benchmark:
H.1.1.a. Extend the properties of exponents to rational exponents.
H.1.1.a.ii. Rewrite expressions involving radicals and rational exponents using the
properties of exponents.
Objectives:


find all the prime factors of a number
simplify radical expressions
Radicals


2
Because 2 = 4 :
2 is a square root of 4
64
In the expression
and 64 is the radicand.
, the
is the radical sign
Example1
Find the square root:
441
2
= 21
= 21
Practice1
Find the square root:
0.04
2
= 0.2
 0.2
Perfect Squares
List some perfect squares.
1
1• 1= ____
4
2•2 = ____
9
3•3 = ____
16
4•4 = _____
25
5•5 = _____
36
6•6 = ____
49
7•7 = ____
64
8•8 = _____
81
9•9 = _____
100
10•10 = ______
The FANCY way to simplify
a radical expression
Simplify :
147
12.1244
Ugly Answer: ____________
The FANCY way to simplify
a radical expression
Simplify :
147
Step 1: Find a perfect square that goes into
147.
49 · 3
The FANCY way to simplify
a radical expression
Simplify :
147
Step 2: Split up the radical and simplify.
49 · 3
49 · 3
=7 3
The EASY way to simplify
a radical expression


Factoring is a way to find ALL the prime numbers that go
into a number.
A prime number can only be divided by 1 and itself.
Example 2
Find all the prime factors of 300.
300
150
2
75
15
3
2
5
5
Practice 2
Find all the prime factors of 126.
126
63
2
21
7
3
3
The EASY way to simplify
a radical expression


When simplifying a square root, the first step is to factor
the number.
Only take 1 number from each pair outside of the radical.
Example 3: Use whiteboard to factor
Simplify the following expression:
605
605
121
5
11
11
Example 3
Simplify the following expression:
605
11 5 ·11·11
11 5
Example 3
Simplify the following expression:
605
DOUBLE CHECK!!
605 =
24.5967
11 5 =
24.5967
Practice 3
Simplify the following expression:
72
72
36
2
18
9
3
2
2
3
Practice 3
Simplify the following expression:
2 ·3
72
2 · 2 · 2 · 3· 3
6 2
Practice 3
Simplify the following expression:
72
DOUBLE CHECK!!
72 =
8.48528
6 2=
8.48528
Example 4
Simplify the following expression:
75
75a b
5 6
15
5
5
3
Example 4
Simplify the following expression:
75a b
5 6
5 · 5 · 3· aaaaabbbbbb
5 ·a ·a ·b ·b·b · 3a
5a b
2 3
3a
Practice 4
Simplify the following expression:
90
90x y
4 3
45
2
9
3
5
3
Practice 4
Simplify the following expression:
90x y
4 3
2 · 3· 3· 5 · xxxxyyy
3 ·x ·x ·y · 2 · 5y
3x y 10y
2
Example 5
Simplify the following expression:
49
7
2 49a b
4 2
7
Example 5
Simplify the following expression:
2 49a b
2 7  7  aaaabb
2·7 ·a·a ·b
14a b
2
4 2
Practice 5
Simplify the following expression:
8
3 8x
4
2
2
2
9
Practice 5
Simplify the following expression:
3 8x
9
3 2 · 2 · 2 · x = 3 2 · 2 · 2xxxxxxxxx =
9
3·2 ·x·x ·x ·x 2x
6x
4
2x
Example 6
The formula s = a can be used to find the
size of a square sector where a is the area of
the square and s is the length of a side. What
is the length of the side of a square with an
area of 729 sq. miles?
729 = a or s?
27
miles
729 =
Whiteboards!!!
Find the square root:
36
 6
2
6
Find the square root:
81
 9
2
9
Find the square root:
49
 7
2
7
Draw a factor tree for:
50
50
5
10
5
2
Simplify the following expression:
5
50
55 2
5 2
Draw a factor tree for:
63
63
7
9
3
3
Simplify the following expression:
3
63
33 7
3 7