Cube Numbers

Cube numbers are numbers obtained by multiplying a whole number thrice. Exploring cube numbers deepens one’s understanding of arithmetic and number patterns and lays a foundation for more advanced mathematical concepts. 

What are Cube Numbers?

Cube numbers involve raising a number to the power of three. When a whole number is multiplied thrice, the resulting value is known as a cube number. The symbol for cubed is 3.

If we have an integer “n,” its cube is represented as “n3,” and it can be calculated as n × n × n

For example, if we take the number 2 and cube it, we get 23 , which equals 8. Similarly, the cube of 3 (33) is 27, and the cube of 4 (43) is 64, and so on.

Cube numbers can be both positive and negative. Whether a cube number is positive or negative depends on the cubed original number.  we cube a negative number and raise it to an odd exponent (3).

In mathematics, an odd exponent preserves the sign of the base number. If the base number is negative, the result will also be negative.

Example: (-6)3 = – 6 × – 6 × – 6 = – 216 (three negative signs multiplied by each other, resulting in a negative sign). 

In contrast, the result would be positive if you raise a negative number to an even exponent, such as squaring it (raising it to the power of 2). This is because squaring an integer eliminates the negative sign due to the even exponent.

Cube Numbers Facts

1. When even numbers are cubed, the result remains even.

Example:

• The cube of 2:  23 = 8 (both 2 and 8 are even)
• The cube of 4:  43 = 64 (both 4 and 64 are even)

1. When odd numbers are cubed, the result will always be odd.

Example:

• The cube of 1: 13 = 1 (both 1 and the result 1 are odd)
• The cube of 3: 33 = 27 (both 3 and the result 27 are odd)

1. The last digit of the cube remains the same as the last digit of the original number. 

Example: The cube of number 4 is 64, a cube of number 5 is 125

However, there are exceptions for the numbers 2, 8, 3, and 27. 

Example: The cube of number 2 is 8 and the cube of number 3 is 27 similarly the cube of number 8 is 512, and cube of number 7 is 343

1. Perfect cubes can be expressed as a sum of consecutive odd numbers.

Example: The cube of 2 is 8 and can be represented as 3 + 5 = 8 (sum of two consecutive odd numbers).

The cube of 3 is 27 can be represented as  7 + 9 + 11 = 27 (sum of three consecutive odd numbers).

1. The cube of a positive number is positive, and the cube of a negative number is negative. This is because raising a negative number to an odd power maintains its negative sign.

Question

 1: Calculate the cube of 5.

 2: List the first four cube numbers.

3. 2197 is the cube of which number?

 4: Express 64 as a sum of consecutive odd numbers.

Solutions

 1: The cube of 5 is calculated by raising 5 to the power of 3.

 53 = 5 × 5 × 5 = 125

 2: The first four cube numbers are: 

 13 = 1 

 23 = 8

 33 = 27 

 43 = 64

3: 2197 is the cube of 13

 13 x 13 x 13 = 2197

4: Let’s find consecutive odd numbers that sum up to 64

1 + 3 + 5 + 7 + 9 + 11 + 13 + 15 = 64

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