Squares and Square Roots
The Square
 Has 4 sides or equal length
 Area can be found by:
length x width
or
length x length
All sides are the same length.
or
length2
Anything multiplied by itself, we use
exponents
Definitions
Square:
Refers to the area of the square
The product of the side lengths multiplying together
Square Root:
Refers to the side length of the square
The number that multiplies by itself to get the area of the square
Ex.
Square
Area
(units2)
Square Root
Side Length
(unit)
Perfect Square:
A square whose side lengths are whole numbers
Ex.
Area: 81 cm2
Side Length
= 9 cm
Area:
5 cm2
Non-perfect Square
Perfect Square
Side Length ≈ 2.2 cm
Not a whole number
Finding Square Roots of Perfect Squares
Benchmarks
Square
Root
(side
length)
1
2
3
4
5
6
7
8
9
10
11
12
Square
(area)
1
4
9
16
25
36
49
64
81
100
121
144
Prime Factorization
Ex. 225
Prime
Factors
3
3
5
5
225
75
25
5
1
3 3
3
1. Find the prime factors of the number
a. use a factor tree
b. or use the chart method
2. Split the prime factors into paired groups.
a. If there are no paired groups or uneven numbers of prime factors then the
number is NOT a perfect Square
5 5
x
5 = 15
3. Take one number from each group and multiply them. The result is your square root.
Square Root = 15
Ex. 324
Prime
Factors
2
2
3
3
3
3
2 2
3
Ex. 676
Prime
676
Factors
2 338
2 169
13 13
13 1
2 2
13 13
324
162
81
27
9
3
1
3 3 3
2
2
x
3 x
3
= 18
x
13 = 26
Ex. 2025
2025
3 675
3 225
3 75
3 25
5 5
5 1
Ex. 625
625
5 125
5 25
5 5
5 1
5 x 5 = 25
3 x 3 x 5 = 45
Ex. 2055
2055
3 685
5 137
Because there are no matching pairs we
know that this is NOT PERFECT
Ex. 1296
1296
2 648
2 324
2 162
2 81
3 27
3 9
3 3
3 1
2 x 2 x 3 x 3 = 36