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**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.

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

a^{3 }

7^{3}

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

27^{3 }= 27 x 27 x 27 = 19638

Let’s take another example

22^{3 }= 22 x 22 x 22 which turns out to be 10648

Have a look at cubes of first 10 natural numbers

1^{3 }= 1 x 1 x 1 = 1

2^{3 }= 2 x 2 x 2 = 8

3^{3 }= 3 x 3 x 3 = 27

4^{3 }= 4 x 4 x 4 = 64

5^{3 }= 5 x 5 x 5 = 125

6^{3 }= 6 x 6 x 6 = 216

7^{3 }= 7 x 7 x 7 = 343

8^{3 }= 8 x 8 x 8 = 512

9^{3 }= 9 x 9 x 9 = 729

10^{3 }= 10 x 10 x 10 = 1000

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

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.

1^{3} + 2 ^{3} + 3^{3} + … + n^{3} is equal to (1 + 2 + 3 + … + n)^{2}

Example: Let’s add cubes of first 5 natural numbers

1^{3} + 2 ^{3} + 3^{3} + 4^{3+ }5^{3}

= 1 + 8+ 27 + 64 + 125

= 225

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

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