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What are Square or Cubic Numbers?

Square Numbers and Square Roots

If a whole number is multiplied by itself (i.e. squared), the result is called a square number. 

For example:

02 = 0              12 = 1              22 = 4              32 = 9              42 = 16            52 = 25

So the numbers 0, 1, 4, 9, 16, and 25 are all square numbers.  Often we think of these numbers as areas of squares, computed by multiplying the length times the width (both dimensions are the same for squares).  If we think of square numbers in this way, then we can define square roots.  Consider the square whose area is 16.  Then its length (and width) would be 4, called the square root of 16.  We use a radical sign to indicate square roots:

= 0            = 1             = 2             = 3              = 4            = 5

We can find square roots of other numbers besides square numbers, but they are irrational numbers and often will need to be approximated.  For example,
≈ 2.236.  Notice that (2.236)2 = 4.999696 or very close to 5.

Cubic Numbers and Cube Roots

If a whole number is raised to the third power (i.e. cubed), the result is called a cubic number.  For example:

03 = 0              13 = 1              23 = 8              33 = 27            43 = 64            53 = 125

So the numbers 0, 1, 8, 27, 64, and 125 are all cubic numbers.  Often we think of these numbers as volumes of cubes, whose length, width, and height are all equal.  If we think of cubic numbers in this way, then we can define cube roots.  Consider the cube whose volume is 27.  Then its length, width, and height would all equal 3, called the cube root of 27.  We use a radical sign with the number 3, to denote cube roots:

= 0            = 1             = 2             = 3          = 4          = 5

We can find cube roots of other numbers besides cubic numbers, but they are irrational numbers and often will need to be approximated.  For example, ≈1.710.  Notice that (1.710)3 = 5.000211 or very close to 5.

These notations and symbols can be extended to all real numbers.  See the tutorial on Exponential Functions.