3.2 Perfect Squares, Perfect Cubes and their Roots
Perfect Squares
The square root of a number n, denoted
49 =
7x 7 =7
n , is a positive number whose square is n.
**It is convenient to think of square root as the side
Length of a square whose area is 49 squared cm.
Area = 49 cm2 7 cm
7 cm
List of perfect squares we should be familiar with:
1, 4, 9, ______________________________________________________
Perfect Cubes
The cube root of a number n, denoted
3
125 =
3
n , is a positive number whose cube is n.
5x 5x 5 = 5 **It is convenient to think of cube root as the side
length of a cube whose volume is 125 cubic cm
125 cm3
5cm
5 cm
5 cm
List of perfect cubes we should be familiar with:
1, 8, 27, ______________________________________
Using Prime Factorization to find Square Roots and Cube Roots without the
aid of a Calculator
Example 1 : Find
72900 without using a calculator
Step 1: Find the Prime Factorization of 72 900
72 900
729 100
9 81
10 10
3 3 9 9 2 5 2 5
3 3 3 3 3 3 2 5 2 5
Step 2: Arrange all the factors of the prime factorization into two equal groups. If
it cannot be done, then 72 900 is not a square number.
72 900
=
(2 3 3 3 5) ( 2 3 3 3 5)
Step 3: Determine the value of each group of factors from step 2. That is the
square root of 72 900.
2 3 3 3 5 = 270
Therefore
Now you try : Determine
72900 = 270. No calculator was needed
396900 without the aid of a calculator
Example 2: Find
3
5832000 without using a calculator
Step 1 : Find the Prime Factorization of 5 832 000
5 832 000
5832
1000
Step 2 : Arrange the factors of the prime factorization into three equal groups. If it
cannot be done, then 5 832 000 is not a cube number
Step 3: Determine the value of each group. That is the cube root of 5 832 000
Now you try: Determine 3 1728000 without the aid of a calculator
Last task: Determine if 46656 a square number or a cube number