# Cube Numbers – Definition, Examples, Facts, Practice Problems

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## What Are Cube Numbers?

We know that a “number squared” refers to the product of the number with itself. So, to find a square, we simply multiply the number with itself once.

What is a cubed number? A “number cubed” refers to the multiplication of the number with itself and again by itself, i.e., multiplication of the same number two times.

In other words, we can find the cube of a number by multiplying the square of a number by the number itself.

A cube number is obtained by raising a given number to the power of 3.

Thus, for a given number n, the cube number is “n cubed” or “$n^3$.”

### Cube Numbers Definition

A cube number is the result of the product of three identical numbers.

A cube number is the number raised to the power of 3.

A cube number is the result of multiplication of a number with itself twice.

n cubed $= n^3 = n \times n \times n$

Cube numbers of positive numbers are positive.

Cube numbers of negative numbers are negative.

Examples of cube numbers:

$(-2)^3 = (-2)\times(-2)\times(-2) =\;-8$

$6^3 = 6\times6\times6 = 216$

$10^3 = 10\times10\times10 = 1000$

$(-10)^3 = (-10)\times(-10)\times(-10) = -1000$

## Why Are They Called Cube Numbers?

The best example of a cube number in geometry is the volume of a cube.

A cube is a three-dimensional shape or a geometric solid. All its sides are equal in length.

The volume of a cube with side “s” is given by $s^3$ or “side cubed.”

Volume of a cube with side 1 unit $= 1^3 = 1$

Volume of a cube with side 2 units $= 2^3 = 8$

Volume of a cube with side 3 units $= 3^3 = 27$

and so on.

Since a cube is a three-dimensional shape with the same length, width, and height of its sides, its volume can be calculated by “cubing” its length.

A cube with side-length 1 unit is called a unit cube. If “n” such unit cubes perfectly fit inside a given cube, then we say that the volume of a given cube is “n cubic units.”

## How to Cube Numbers

To cube numbers, simply multiply the square of a number by itself.

The standard form of a cube of any number “a” is “$a^3$,” which can be obtained using $a^3 = a\times a \times a$. This applies to every number.

For example, $3 \times 3 \times 3 = 27,\; 8 \times 8 \times 8 = 512,\; 10 \times 10 \times 10 = 1000$, etc.

## Cube Numbers vs. Square Numbers

The cube numbers are the numbers obtained by raising a number to the power of 3, whereas a square is obtained by multiplying a number by itself.

Square of $n = n^2 = n \times n$  …a number multiplied by itself once

Cube of $n = n^3 = n \times n \times n$ …a number multiplied by itself twice

Let us consider an example.

Multiply 4 by itself to get its square.

4 squared $= 4^2 = 4\times4 = 16$ …4 multiplied by itself once

Multiplying 42 by 4, we get

4 cubed $= 4^3 = 4\times4\times4 = 64$ …4 multiplied by itself twice

Here, 64 is a cube number.

## Cube Numbers in Real Life

While cube numbers are excellent for assisting students in comprehending patterns and relationships between numbers, they also have practical applications. Let’s say we want to know the volume of a cubic object like a container; all we need to do is cube the value of one of its sides.

To determine the dimensions of gift boxes, size of ice cubes, etc., are a few of the many real-life uses for cube numbers.

• The last digits of a cube number are 1, 2, 3, 4, 6, 7, 8, 9, 000, and 125, 375, 625, or 875.
• The Rubik’s Cube is a cube of number 3.
• The result of the cube of even numbers will be an even number. For example, $4^3 = 64,\; 6^3 = 216$, etc.
• The result of the cube of odd numbers will be an odd number. For example, $3^3 = 27,\; 5^3 = 125$, etc.
• The sum of consecutive odd numbers can be used to describe perfect cube numbers. For example, 27 is a cube number of 3, and it can be written as the sum of the consecutive odd numbers as $7 + 9 + 11 = 27$, similarly $64 = 1 + 3 + 5 + 7 + 9 + 11 + 13 + 15$, etc.
• A cube number represents the volume of a cube.

## Conclusion

In this article, we have learned about cube numbers, their visualization, list, and examples. Let’s solve a few examples and practice problems.

## Solved Examples on Cube Numbers

1. Find out the cube of numbers $9,\; -15$, and 19.

Solution:

Cube of numbers 9, 15, and 19 calculated as

$9^3 = 9 \times 9 \times 9 = 929$

$(-15)^3 = (-15) \times (-15) \times (-15) = 3375$

$19^3 = 19 \times 19 \times 19 = 6859$

1. Evaluate $25^3$.

Solution:

$25^3 = 25 \times 25 \times 25$

$= 625 \times 25$

$= 15625$

1. Write the cube number 64 and 125 as the sum of the consecutive odd numbers.

Solution:

64 and 125 as the sum of the consecutive odd numbers can be written as

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

$125 = 21 + 23 + 25 + 27 + 29$.

1. If the side length of a cube is 12 units, find its volume.

Solution:

The volume of a cube can be calculated by “cubing” its side length.

The side length of a cube $= 12$ units

$\therefore$ Volume $= 12^3 = 12 \times 12 \times 12$

$= 1728$ cubic units.

1. Explain why the number 512 is a cube number.

Solution:

The number 512 can be written as $512 = 8 \times 8 \times 8$.

Here, the result of multiplying the number 8 by itself and then by itself again is 512.

And a cube number is defined as the result of multiplying a number by itself and then by itself again. Thus, 512 is a cube number.

## Practice Problems on Cube Numbers

1

### Which of the following is a cube number?

9
16
27
36
CorrectIncorrect
$3 \times 3 \times 3 = 27$, hence 27 is a cube number.
2

### $3^3 + 6^3 =$ ______.

45
143
243
343
CorrectIncorrect
$3^3 = 3 \times 3 \times 3 = 27, 6^3 = 6 \times 6 \times 6 = 216$
Therefore, $3^3 + 6^3 = 27 + 216 = 243$
3

### A perfect cube can be expressed as the ____ of a cube.

volume
base area
lateral surface area
Total surface area
CorrectIncorrect
A perfect cube can be expressed as the volume of a cube.
4

### The cube of a negative number is _______.

positive
negative
both a and b
0
CorrectIncorrect
The cube of a negative number is negative.

## Frequently Asked Questions on Cube Numbers

Yes, 343 is a cube number as it can be written as $343^3 = 7 \times 7 \times 7$.

Yes, cubes of negative numbers are negative cube numbers. For example, $(−2)^3 = −8,\; (−3)^3 = −27,\; (−4)^3 = −64$, etc.

Cube of $−5 = (−5)^3 = −125$.

Perfect cube numbers between 1 and 500 are 1, 8, 27, 64, 125, 216, and 343.

Hence, there are a total of eight perfect cube numbers between 1 and 500.

Perfect cube numbers between 1 and 100 are 1, 8, 27, and 64, out of which only 1 and 64 are perfect squares.

1 cubed $= 1^3 = 1$