1. Cube Numbers

Cubes and Cube Roots

Cube Numbers

Cube number

a cube is a 3-dimensional figure.  When we multiply a number by itself and then by itself again (thrice), the product is a cube number. It is also called as a perfect cube. That is, if a is a number, its cube is represented by a ^3.

A cube is a solid figure, which has all sides of equal length.

The following table consist of cube numbers of the first ten numbers. 

 Properties of cube numbers

1. The cube of a positive number is always positive.

Example: 4³=4×4×4=64

 2. The cube of a negative number is always negative.

Example: (−4)³=(−4)×(−4)×(−4)=−64

 3. The cube of every even number is even.

Example: 2³=8, 4³=64, 6³=216, 8³=512, ...

 Here, 8, 64, 216 and 512 are all even numbers.

 4. The cube of every odd number is odd.

Example: 1³=1, 3³=27, 5³=125, 7³=343, ...

Here, 1, 27, 2125 and 343 are all odd numbers.

5. If a natural number ends at 0, 1, 4, 5, 6 or 9, its cube also ends with the same 0, 1, 4, 5, 6 or 9, respectively.

Example: (i) 10=1000

6. If a natural number ends at 2 or 8, its cube ends at 8 or 2, respectively.

Example: (i) 2³=8–           

                 (ii) 8³=512–

7. If a natural number ends at 3 or 7, its cube ends at 7 or 3, respectively.

Example: (i) 3³=27

                 (ii) 7³=343

8. A perfect cube does not end with two zeroes.

Example: 10³=1000, 20³=8000, …

9. The sum of the cubes of first n natural numbers is equal to the square of their sum.

That is, 1³+2³+3³+4³+….+n³=(1+2+3+4+…+n)²

Example: 1³+2³+3³=1+8+27=36

                         (1+2+3)²=6²=36

             So, 1³+2³+3³=(1+2+3)²

10. Each prime factor of a number appears three times in its cube.

Example:  6³=216

Prime factor of 6 = 2×3

Prime factor of 216 = (2×2×2)×(3×3×3)

 11. There are only three numbers whose cube is equal to itself.

 (i) 0³=0×0×0=0

 (ii) 1³=1×1×1=1

 (iii) (−1)³=(−1)×(−1)×(−1)=−1

Example problems for cube numbers

1. Find the cube of 21.

 Solution:

 If you multiply a number by itself and then by itself again (thrice), the result is a cube number.

 21=21×21×21=9261

 Therefore, the cube of 21 is 9261.

2. Is 243 is a perfect cube? If not, find the smallest number to divide 243 to make it as a perfect cube.

 Solution:

 243=(3×3×3)×3×3

 Here, factor 3×3=9 is leftover while grouping.

 So, 243 is not a perfect cube.

 To make it as a perfect cube, divide the given number by the leftover factor 9.

2439=27

 Therefore, 9 is the smallest number to divide 243, and the obtained perfect square is 27.