Numbers
LESSON 2: PERFECT SQUARES AND CUBES,
SQUARE AND CUBE ROOTS
Todays Objectives
 Students will be able to demonstrate an
understanding of factors of whole numbers by
determining the: prime factors, greatest common
factor (GCF), least common multiple (LCM),
square root, cube root, including:
Determine, concretely, whether a given whole number is
a perfect square, a perfect cube or neither
 Determine, using a variety of strategies, the square root
of a perfect square, and explain the process
 Determine, using a variety of strategies, the cube root of a
perfect cube, and explain the process

Perfect Squares and Perfect Cubes
 For whole numbers, the word perfect in the term
perfect square means that the square can be written as a
product of two identical whole numbers.
 Similarly, a perfect cube can be written as a product of
three identical whole numbers.
 For example, 81 is a perfect square because it can be
written as 9 x 9, and 8 is a perfect cube because it can be
written as 2 x 2 x 2.
 A perfect square can be represented as the area of a
square with whole number dimensions. Similarly, a
perfect cube can be represented as the volume of a cube
with whole number dimensions.
Perfect Squares and Perfect Cubes
2 cm
2 cm
Area = 4
square
cm (cm2)
2 cm
Volume =
8 cubic cm
(cm3)
2 cm
2 cm
Perfect Square
Perfect Cube
Perfect Squares and Perfect Cubes
 If you have square tiles, you could use
them to determine or show that a number
is a perfect square by actually forming a
square with the tiles.
 If you have some small cubes, such as
dice, you can use them to determine or
show that a number is a perfect cube by
constructing a larger cube out of the
smaller ones.
Example
 How could you use 36 square tiles to show that the
number 36 is a perfect square?
 Solution: You could construct a square using all 36
tiles. The resulting square will have dimensions 6
tiles x 6 tiles. Thus, 6 x 6 = 36, and 36 is a perfect
square
 How could you use 30 sugar cubes to show that the
number 30 is NOT a perfect cube?
 Solution: You could attempt to construct a cube out
of the sugar cubes and prove that it is impossible,
therefor proving that 30 is NOT a perfect cube.
Square Roots and Cube Roots
 The square root of a number, x, is the number, s, such
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that s x s = x.
For example, the square root of 4, is 2 (2 x 2 = 4)
The cube root of a number, x, is the number q, such that
q x q x q = x.
For example, the cube root of 8 is 2 (2 x 2 x 2 = 8)
You should be able to recognize perfect squares and cubes
that are 100 or less, and, consequently, know their square
and cube roots.
For example, you should be able to recognize 49 as a
perfect square (49 = 7 x 7) and 27 as a perfect cube (3 x 3
x 3 = 27).
Knowing these allows you to state that the positive square
root of 49 is 7, and the cube root of 27 is 3.
Positive and Negative Roots
 The symbol that we use to show the operation of taking
the positive square root is √, and the symbol for cube
root is 3√. These symbols are called radical signs.
 The symbol √ represents the positive square root so that
√49 = 7. The negative square root is represented by -√
so that, for example, -√49 = -(7) = -7.
 The symbol for both the positive and negative square
roots is ±√ so that, for example, ±√49 = ±7.
 The square root of a negative number is not a real
number so that, for example, √-49 does not exist as a
real number. There are however cube roots of positive
and negative numbers so that , for example, 3√27 = 3
and 3√-27 = -3.
Square Roots of Perfect Squares and Cube Roots
of Perfect Cubes
 You should be able to recognize perfect squares and
perfect cubes that are relatively small numbers, such
as the ones shown in the following chart.
Perfect Squares
Perfect Cubes
1 = 12
25 = 52
81 = 92
8 = 23
4 = 22
36 = 62
100 = 102
27 = 33
9 = 32
49 = 72
121 = 112
125 = 53
16 = 42
64 = 82
144 = 122
1000 = 103
 Being able to recognize basic perfect squares and
perfect cubes, can help you find or estimate square
and cube roots without using a calculator.
Example
Determine the positive square root of 3969 without using a
calculator
 Solution: First, recognize that 6 x 6 = 36, so that 60 x 60 =
3600, which is smaller than 3969.
 Also, 70 x 70 = 4900, which is larger than 3969. Thus, the
square root of 3969 is between 60 and 70, and likely closer to
60, because 3600 is closer to 3969 when compared to 4900.
 Notice that the ones digit in 3969 is 9, and therefore, if 3969
is a perfect square, the ones digit in its square root must be a 3
or 7. Why?
 3 and 7 are the only single-digit numbers whose squares have
a ones digit of 9. Thus, try 63 as the possible square root by
multiplying it by itself:
 63 x 63 = 3969, therefore, √3969 is 63.
Example (You do)
Determine the positive square root of 1156 without
using a calculator, and explain the process you used.
 Solution:
 3 x 3 = 9, so 30 x 30 = 900. Also, 4 x 4 = 16, so 40 x
40 = 1600.
 Answer is likely closer to 30, because 900 is closer to
1156 than 1600.
 The ones digit is 6, so ones digit of square root must
be 4 or 6….try 34 and 36.
 34 x 34 = 1156, therefore, √1156 = 34.
Example (You do)
Determine the cube root of 12 167 without using a
calculator.
 Solution:
 103 = 1000, 203 = 8000, 303 = 27000, so the cube
root of 12167 is between 20 and 30, and closer to 20.
 The ones digit is 7, so if 12167 is a perfect cube, the
ones digit must be a 3. So try 23 as the cube root.
 23 x 23 = 529
 529 x 23 = 12167
 Therefore, 3√12167 = 23.
Wall Quiz!
 Teams of 3
 When I say go, your team should move around to the
different questions located on the walls
 You cannot bring a calculator!
 Try to answer each question, and record your
answers on a piece of paper
 The team that gets the most questions right in the
given time limit will win candy!