Cube Numbers(Meaning, Formulas & Examples)

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In geometry, a cube is a 3D solid object with six square faces and all the sides of a cube are of the same length. The volume of a cube is computed using the formula $a \times a \times a = a^{3}$, where $a$ is the length of the edge(or side) of a cube. Similarly, with reference to numbers, a cube number is a number whose value is equal to a number multiplied by itself two more times, ie., $a \times \left(a \times a \right)$.

Let’s understand what is a cube number, how you find the cube number, and what are the properties of a cube number.

What are Cube Numbers?

A cube number is a result when a number is multiplied by itself twice. Basically, a cube number is a number obtained by the product of three same numbers. Mathematically, a cube number of a number $a$ is $a \times a \times a = a^{3}$.

In geometry, the volume of a cube is the finest example of a cube number.

Cube Numbers of Odd and Even Numbers

When you multiply an odd number by itself any number of times, you always get an odd number, i.e., if $a$ is an odd number, then $a \times a \times a \times … \text{ n times } = a^{n}$ is always an odd number and hence, $a^{3}$ is an odd number. 

Thus, the cube number of an odd number is always an odd number.

For example, $11$ is an odd number and $11^{3} = 11 \times 11 \times 11 = 1331$ is also an odd number.

When you multiply an even number by itself any number of times, you always get an even number, i.e., if $a$ is an even number, then $a \times a \times a \times … \text{ n times } = a^{n}$ is always an even number and hence, $a^{3}$ is an even number. 

Thus, the cube number of an even number is always an even number.

For example, $8$ is an even number and $8^{3} = 8 \times 8 \times 8 = 512$ is also an even number.

How To Find Cube of a Number?

If you get your cubes and roots mixed up, keep in mind that cubing a number is as simple as multiplying it by itself. Because of this, it’s important to know how to multiply single digits as well as larger numbers. To cube fractions, find the cubes of both the numerator and denominator. Then reduce or simplify the result.

Following are the procedures to get a cube of numbers.

Cube Numbers of Numbers From 1 To 10

When you square a number, you simply multiply the number by itself so it’s important to know how to multiply. For example, the cube of $3$ is $3^{3} = 3 \times 3 \times 3 = 27$. To make it easier to cube commonly used single digits, try to memorize basic multiplication tables.

$1 = 1 \times 1 \times 1 = 1^{3}$

$8 = 2 \times 2 \times 2 = 2^{3}$

$27 = 3 \times 3 \times 3 = 3^{3}$

$64 = 4 \times 4 \times 4 = 4^{3}$

$125 = 5 \times 5 \times 5 = 5^{3}$

$216 = 6 \times 6 \times 6 = 6^{3}$

$343 = 7 \times 7 \times 7 = 7^{3}$

$512 = 8 \times 8 \times 8 = 8^{3}$

$729 = 9 \times 9 \times 9 = 9^{3}$

$1000 = 10 \times 10 \times 10 = 10^{3}$

Cube of Larger Numbers

The basic process of finding the cube of large numbers also remains the same, i.e., multiply the number by itself two more times. You can use certain tricks or properties of numbers to find the cube number of larger numbers.

Method 1: Multiplication By Itself

In this method, the number is multiplied by itself two more times and the resultant product gives us the cube number of that number. For example, the cube of $16^{3} = 16 \times 16 \times 16 = 4096$. Here, the resultant product $4096$ gives us the cube number of the number $16$ This method works well for smaller numbers.

Method 2: Using Basic Algebraic Identities

When the numbers are large, you can use algebraic identities to find the cube of a number. The two algebraic identities that are helpful here are

• $\left(a + b \right)^{3} = a^{3} + 2a^{2}b + 2ab^{2} + b^{3}$
• $\left(a – b \right)^{3} = a^{3} – 2a^{2}b + 2ab^{2} – b^{3}$

Examples

Let’s consider some examples to find a cube number of numbers using these identities.

Ex 1: Cube number of $23$

Cube of $23$ is $23^{3} = \left(20 + 3\right)^{3}$

Observe that $ \left(20 + 3\right)^{3}$ is in the form  $\left(a + b\right)^{3}$, where $a = 20$ and $b = 3$.

Therefore, $ \left(20 + 3\right)^{3}$ becomes $20^{3} + 3 \times 20^{2} \times 3 + 3 \times 20 \times 3^{2} + 3^{3} = 8000 + 3600 + 540 + 27 = 12167$

Cube number of $23$ can also be calculated using the identity $\left(a – b \right)^{3} = a^{3} – 3a^{2}b + 3ab^{2} + b^{3}$

$23^{2}$ can be written as $\left(30 – 7 \right)^{2}$, where $a = 30$ and $b = 7$.

Therefore, $\left(30 – 7 \right)^{3} = 30^{3} – 3 \times 30^{2} \times 7 + 3 \times 30 \times 7^{2} – 7^{3} = 27000 – 18900 + 4410 – 343 = 12167$

Thus, we see that the cube number of $23$, i.e., $23^{3}$ is $12167$ using any of the two methods.

Patterns Formed by Cube Numbers

You can observe some interesting patterns followed by cube numbers.

Adding Consecutive Odd Numbers

The odd numbers are the numbers that are not divisible by $2$, i.e., the numbers that leave the remainder $1$ when divided by $2$.

The first few odd numbers are $1$, $3$, $5$, $7$, $9$, $11$, $13$, $15$, $17$, $19$, $21$, …

$1 = 1 = 1^{3}$

$3 + 5 = 8 = 2^{3}$

$7 + 9 + 11 = 27 = 3^{3}$

$13 + 15 + 17 + 19 = 64 = 4^{3}$

$21 + 23 + 25 + 27 + 29 = 125 = 5^{3}$

$31 + 33 + 35 + 37 + 39 + 41 = 216 = 6^{3}$

Properties of Cube Numbers

The cube numbers exhibit the following properties

Property 1: There will be no change in the last digit of a number and the last digit of the cube number of the number, except $2$ will change to $8$ and $8$ will change to $2$ and $3$ will change to $7$, and $7$ will change to $3$. For example, the cube of number $2$ is $8$ and the cube of number $3$ is $27$, and similarly, the cube of number $8$ is $512$, and the cube of number $7$ is $343$, and so on. Let’s take an example of numbers other than $2$, $3$, $7$, and $8$. The cube of the number $4$ is $64$, the cube of the number $5$ is $125$, the cube of the number $14$ is $2744$, and so on. We can see that the last digits of a cube number and the number which is cubed are the same.

Property 2: When even numbers are cubed, the result will be an even number. For example, $2^{3} = 8$, $4^{3} = 64$.

Property 3: When odd numbers are cubed, the result will be an odd number. For example, $3^{3} = 27$, $5^{3} = 125$.

Perfect cube numbers can be both positive and negative integers. For example, both $8$ and $- 8$ are the perfect cubes of $2$ and $- 2$ respectively.

Property 4: Perfect cubes can be expressed as a sum of consecutive odd numbers. For example, $8$ is a cube number of $2$ and it can be expressed as a sum of consecutive odd numbers $3 + 5 = 8$; $27$ is a cube number of $3$ and it can be expressed as a sum of consecutive odd numbers $7 + 9 + 11 = 27$.

What is Perfect Cube?

Perfect cubes are numbers that are the triple product of the same number. In other words, a perfect cube is a value that is the result of three times the multiplication of a whole number to itself. A perfect cube is a number, that can be written as three times the product of the same number.

For example cube of $7$ is $7^{3} = 7 \times 7 \times 7 = 343$. Therefore, $343$ is a perfect cube number.

Let’s consider some examples to understand perfect cube numbers.

Examples

Ex 1: Is $165375$ a perfect cube? If not, then by which smallest natural number should $165375$ be multiplied so that the product is a perfect cube?

The prime factorization of 165375 is $3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7 = 3^{3} \times 5^{3} \times 7^{2}$.

The prime factorization of $165375$  contains perfect cubes of $3$ and $5$, but $7$ is only $2$ times. To include a perfect cube of $7$ in the number, therefore, we need to multiply it by $7$.

Thus, $7$ needs to be multiplied by $165375$ to get a perfect cube.

$165375 \times 7 = 1157625 = 3 \times 3 \times 3 \times 5 \times 5 \times 5 \times 7 \times 7 \times 7 = 3^{3} \times 5^{3} \times 7^{3} = \left(3 \times 5 \times 7 \right)^{3} = 105^{3}$ is a perfect cube number.

Ex 2: Is $1080$ a perfect cube? If not, then by which smallest natural number should $1080$ be divided so that the quotient is a perfect cube?

The prime factorization of $1080$ is $2 \times 2 \times 2 \times 3 \times 3 \times 3 \times 5 = 2^{3} \times 3^{3} \times 5$.

The prime factorization of $1080$  contains perfect cubes of $2$ and $3$, but $5$ is only $1$ times. To have all the prime factors as perfect cubes in the number, we need to remove $5$, i.e., divide the number by $5$.

Thus, $5$ needs to be divided by $1080$ to get a perfect cube.

$1080 \div 5 = 216 = 2 \times 2 \times 2 \times 3 \times 3 \times 3 = 2^{3} \times 3^{3} = \left(2 \times 3 \right)^{3} = 6^{3}$ is a perfect cube number.

Practice Problems

1. Find the cube number of the following numbers using an algebraic identity.
   • $32$
   • $49$
   • $65$
   • $74$
   • $98$
2. Check whether the following are perfect cube numbers or not. If not, then what number should be multiplied to make the product a perfect cube number?
   • $2916$
   • $189$
   • $5400$
3. Check whether the following are perfect cube numbers or not. If not, then what number should be divided to make the quotient a perfect cube number?
   • $16384$
   • $5184$
   • $27648$

FAQs

Conclusion

A cube number is a result when a number is multiplied by itself twice. Basically, a cube number is a number obtained by the product of three same numbers. It means every cube number is also called a perfect cube number. The cube numbers exhibit very interesting patterns.

Recommended Reading

• Square Root of a Number(Meaning, Formulas & Examples)
• What Are Square Numbers(Meaning, Formulas, and Examples)
• What is Percentage – Meaning, Formula & Examples
• What is Proportion? (With Meaning & Examples)
• What is Ratio(Meaning, Simplification & Examples)
• Factors and Multiples (With Methods & Examples)
• Fractions On Number Line – Representation & Examples
• Reducing Fractions – Lowest Form of A Fraction
• Comparing Fractions (With Methods & Examples)
• Like and Unlike Fractions
• Improper Fractions(Definition, Conversions & Examples)
• How To Find Equivalent Fractions? (With Examples)
• 6 Types of Fractions (With Definition, Examples & Uses)
• What is Fraction?  – Definition, Examples & Types
• Mixed Fractions – Definition & Operations (With Examples)
• Multiplication and Division of Fractions
• Addition and Subtraction of Fractions (With Pictures)