08 September 2015
Warm-Up
Solve each equation
and
check the solution.
1) 4  2
(z) 9
2) 5  x1
6 24
3) r  2  5
18 9
4) -4  8
11 x 1
Squares and Square Roots
Definitions:
Square: Multiplying a number, variable or
expression by itself.
Square Root: A number, variable or
expression that is a factor when multiplied by
itself.
Examples
Find the square root of the following:
25  5
144  12
0.81  0.9
1.21  1.1
100  NO Solution
Squares and Square Roots
There is a HUGE difference between using a
calculator to find the value of the square root
and finding the EXACT value of a square root.
Perfect Square: Any number that is created
by multiplying a rational number by itself.
 5  5   25
25  5
(1.2)(1.2)  1.44
1.44  1.2
 0.09  0.09   0.0081
0.0081  0.09
Squares and Square Roots
Sometimes a PERFECT SQUARE is not so
obvious. What is the Square Root of 0.111?
Since 0.111 is a repeating decimal, you will need to
recognize that is a rational number equivalent.
1
0.111 
9
1
0.111 
9
1
1 1


9
9 3
1
 0.333
3
So, in fact 0.111 is a perfect square.
Squares and Square Roots
Prime Numbers: Any number whose only
factors are 1 and itself.
Prime Factorization: Reducing any number
to the product of Prime Numbers.
5 is a Prime Number because you can only multiply
1 and 5 to get the product 5.
Prime Numbers: 2, 3, 5, 7, 11, 13, 17 …..
NOTE:
There are infinitely many Prime Numbers.
12 is not a Prime Number because it has more
factors than just 12 and 1.
3  4  12
2  6  12
2  2  3  12
Squares and Square Roots
Look at this product that equals 12 in particular
2  2  3  12
Since the numbers 2 and 3 are Prime Numbers,
this product represent the Prime Factorization of
12. You will need to use the Prime Factorization
of numbers to find the EXACT value a number
that is NOT a Perfect Square.
If you use a calculator to find the APPROXIMATE
value of 12, you will get the following.
12  3.464101615...  3.464
The best you are likely to do is get a number
rounded to 2 or 3 decimal places. The EXACT
VALUE will have a Square Root term in the
answer.
12 
 2 23  2
3
Squares and Square Roots
Handy Rules for Prime Factorization
1) If the number is even then 2 is a factor.
2) If the sum of the digits in the number add
up to a multiple of 3 then 3 is a factor.
3) If the number ends with a 5 or a 0 then 5
is a factor.
EXAMPLE: 540 ends in a 0 so 5 is a factor, it
is even so 2 is a factor and the digits add to 9 so 3
is a factor.
5  108   540
5   3  36   540
5*3*  3*12   540
5  3  3   3  4   540
5  3  3  3  2  2  540
Squares and Square Roots
Now, to find the EXACT value of the Square
Root of 540 use the Prime Factorization.
5  3  3  3  2  2  540
3  2  5  3  540
6 15  540
Any Questions?
Squares and Square Roots
There are two important rules you need to
remember.
Product Property of Square Roots:
ab  a  b
Quotient Property of Square Roots:
a
a

b
b
Squares and Square Roots
Find the EXACT value and the
APPROXIMATE value of the following.
1) 18
2) 140
3) 147
4) 605
5) 175
September 08, 2015
Homework
Copy and find the exact value and the
approximate value of the following
expressions.
1)
75
2) 80
3)
280
4) 500
5) 169
6) 1.96
7) 729
8) 1089
9) 441
10) 0.01
11) 136
12) 40804