Lesson 4
Math 1201
Perfect Squares, Perfect Cubes, and Their Roots
In Grade 8, perfect square numbers were connected to the area of squares.
When determining the square root of a whole number, we can envision a
square as the perfect square number, and either side length as the square
root.
For example, a square with side length 3 has an area of 9 tiles.
3
9 3
or
32  9
3
Students can use a variety of strategies to determine the square root of a
perfect square.
If a number is the product of two equal factors or if the factors can be
grouped in equivalent sets, the number is a perfect square. For example,
324 is a perfect square since it can be rewritten as 2  3  3  2  3  3,
therefore the value of
324 
2  3  3 2  3  3  2  3  3  18 .
Another method involves whether a number has an odd or even number of
factors. If a number has an odd number of factors, the number is a perfect
square. When the factors are listed in order, the middle factor is defined to
be the square root. For example, 36 has the factors 1,2,3,4,6,9,12,18 and 36.
There are 9 factors, which is an odd number. Therefore, 36 is a perfect
square and 6, the middle factor, is the square root.
When determining the cube root of a whole number, students should view
the perfect cube number as the volume and cube root as any one of the
three equivalent dimensions.
For example, a cube with side length 4 cm has a volume of 64 cm3. We can
verify the value of the volume using the formula V  l  w  h .
New notation, 3 64  4 , means the cube(third) root of sixty four is equal to
four. index is a way to determine what root the questions is referring. For
example,
so on.
4
represents the fourth root,
5
represents the sixth root and
represents square root and the index of 2 is implied and not
written.
Student can use prime factorization to determine the cube root of number.
For example, 3 216  3 2  3  2  3  2  3  6
Worksheet
Math 1201
Name:___________
1.
List the factors of 256. Use the factors to determine the square
root of 256.
2.
If 1000 cubes were combined to make a giant cube, what is the area
of each face?
3.
Simplify the following:
A)
3
B)
C)
343
121  3 216
3
64  3 1000  25