C. Perfect Squares and Perfect Cubes

Square roots and perfect squares are closely related. For example, the number 9 is a perfect square because it is formed by multiplying the factor 3 by itself.

Square Root
a factor that is multiplied by itself to generate a perfect square

Perfect Square
generated when a factor is multiplied by itself

Example 1

Show the relationship between the number 9 and its square root.

The number 9 is a perfect square.

The number 3 is the square root of the number 9.


So, a perfect square is the product of a factor and itself. The identical factor also represents the square root of the perfect square.

There is a recognizable pattern with perfect square numbers and their square roots.

  • 1 is a perfect square because it is the product of 1 × 1, therefore .

  • 4 is a perfect square because it is the product of 2 × 2, therefore .

  • 9 is a perfect square because it is the product of 3 × 3, therefore .

  • 16 is a perfect square because it is the product of 4 × 4, therefore .

Another way to think of perfect squares and square roots is by considering the side length and area of a square.

Example 2

The following square has a side length of 5 units. Determine the area of the square.


Can you see how the area of a square and its side length are related to a perfect square and its square root?