Basic Algebra 1
Chapter 8 Powers and Roots
8-5 Square Roots
 WARMUP
 Express in Scientific Notation:
1. 1,650,000,000
2. 0.00141
Evaluate:
1. (1.5 X 102)(4 X 107)
2. (3.6 X 1011) ÷ (1.2 X 109)
8-5 Square Roots
 GOAL: To learn to simplify radicals by using
the Product and Quotient Properties of
Square Roots…
8-5 Square Roots
 Recall from earlier in the chapter that
squaring a number means using that
number as a factor twice. For example,
x2 = xx
22 = 22 = 4
The opposite of squaring is finding the square
root. To find the square root of a number,
you must find two equal factors whose
product is that number:
66 = 36, so the square root of 36 is 6.
8-5 Square Roots
 Square Root:
 A square root of a number is one of its two
equal factors.

a  b, where a  b  b
8-5 Square Roots
 This symbol:

36  6
 16  4
is called the radical sign.
8-5 Square Roots
 Try some:
 Simplify these:
25
1
0
144
16
8-5 Square Roots
 A radical expression is an expression that
contains a square root.
 Radical expressions can be simplified using
prime factorization…
 We have to recall what a prime number is…
8-5 Square Roots
 A prime number is a whole number that has
exactly two factors, the number itself and the
number 1.
 Which of the following are prime numbers?
2, 4, 5, 7, 8, 9, 13, 15, 17, 20, 21, 23, 27?
8-5 Square Roots
 The first ten prime numbers are:
2, 3, 5, 7, 11, 13, 17, 19, 23, 29
 All non-prime whole numbers are called
composite numbers – whole numbers that
have more than two factors.
 This includes any multiple of any prime
number…
8-5 Square Roots
 The Product Property of Square Roots
 The square root of a product is equal to the
product of each square root.

49  4  9
a b  a  b
8-5 Square Roots
 Now we can get back to prime factorization.
 Let’s do a factor tree for 225
225
3
75
3
3
25
3
3
5
5
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 So, as a result of the previous factor tree,
we know that 225 = 3 X 3 X 5 X 5.
 So we can simplify the following as shown:
225
225  3  3  5  5
 9  25
 9  25
 3  5  15
So, the
square
root of 225
is 15
8-5 Square Roots
 More examples:
144
48
196
8-5 Square Roots
 Quotient Property of Square Roots
 The square root of a quotient is equal to the
quotient of each square root.

4
4

9
9
a
a

b
b
8-5 Square Roots
 More examples
8-5 Square Roots
 HOMEWORK
8-5 Square Roots
8-5 Square Roots
8-5 Square Roots