What is cubed in math?

Book A Demo Class

Cubed Number: The final answer received by multiplying a whole number by itself three times is called cubed number or cube number.

For example: 11 x 11 x 11 = 1331

1331 is called the cube of 11.

Representation

There are two ways to represent a cube number. Let’s check.

Way 1: Expanded Form

Here you write the number in the multiplication form. Let’s check few examples

a x a x a

7 x 7 x 7

Way 2: Superscript Form

For a cube number, represent the number with 3 as superscript. Here are a few examples

a³

7³

How to cube a number?

To find cube of a number

Step 1: First multiply it by itself. This is also known as square of a number.

Step 2: Now again multiply the same number with the result obtained in Step1.

So as per these steps, let’s find the cube of 27.

Step 1: 27 x 27 = 729

Step 2: 729 x 27 = 19638

So, the cube of 27 is 19638. This can also be written as

27³= 27 x 27 x 27 = 19638

Let’s take another example

22³= 22 x 22 x 22 which turns out to be 10648

Have a look at cubes of first 10 natural numbers

1³= 1 x 1 x 1 = 1

2³= 2 x 2 x 2 = 8

3³= 3 x 3 x 3 = 27

4³= 4 x 4 x 4 = 64

5³= 5 x 5 x 5 = 125

6³= 6 x 6 x 6 = 216

7³= 7 x 7 x 7 = 343

8³= 8 x 8 x 8 = 512

9³= 9 x 9 x 9 = 729
10³= 10 x 10 x 10 = 1000

What is a perfect cube?

If a number can be broken into a product of three same numbers, it is called a perfect cube.

Find if a number is a perfect cube or not

Perform prime factorization of the number. If the factors can be grouped into three exact sets of similar numbers, only then it’s a perfect cube.

For example:

1700 = 17 x 10 x 10

1700 is not a perfect cube as it has only two 10’s and one 17.

1728 = 12 x 12 x 12

1728 is a perfect cube as it is made by multiplying 12 three times.

Properties of cube numbers

When an even number is cubed, the answer is also even. For example: 14 x 14 x 14 = 2774. 14 and 2774 both are even numbers.
When an odd number is cubed, the result is also odd. For example: 13 x 13 x 13 =2197. Here, 13 and 2197 both are odd numbers.
The cubes of numbers ending in 1,4, 5, 6 and 9 also have the same number at the unit’s place. For example: In, 14 x 14 x 14 = 2774, 14 and 2774 both end in 4.
If we add the cubes of first n natural numbers, the answer is equal to all the numbers added together and then squared.

1³ + 2 ³ + 3³ + … + n³ is equal to (1 + 2 + 3 + … + n)²

Example: Let’s add cubes of first 5 natural numbers
1³ + 2 ³ + 3³ + 4³⁺5³

= 1 + 8+ 27 + 64 + 125

= 225

This is same as (1 + 2 + 3 + 4 + 5)² = (15)²= 225