Here we will learn about special numbers, Cube and Cube roots. We obtain a cube number by multiplying the number with itself 3 times. Finding Cube Roots are inverse operation of cubing.

Image Credit: Pixabay.com |

# Cube Number

➤ Product we get when a number is multiplied three times by itself is called a **cube number**.

or result we obtain when a number is cubed (raised to power of 3) is called a cube number.

### Perfect Cube

When we multiply an integer three times by itself, cube number we get is called **perfect cube**.

*So, a perfect cube is a cube of an integer*.

(−2)×(−2)×(−2) = −8,

(−1)×(−1)×(−1) = −1,

1×1×1 = 1,

2×2×2 = 8,

3×3×3 = 27, etc

→ *Cube of a negative integer is always a negative integer.*

→ *Cube of a positive integer is always a positive integer.*

➤ Here, we will consider Cubes of only positive integers i.e, natural numbers,

1×1×1 = 1,

2×2×2 = 8,

3×3×3 = 27, etc

## Properties of Cube numbers

Number |
Cube |
Number |
Cube |

1
2 3 4 5 6 7 8 9 10 |
1
8 27 64 125 216 343 512 729 1000 |
11
12 13 14 15 16 17 18 19 20 |
1331
1728 2197 2744 3375 4096 4913 5832 6859 8000 |

→ The cube of an **even** number is always an **even** number and the cube of a **odd** number is always a **odd** number.

### Comparing Unit digits of a Number and its Cube

→ Cube of a number ending in digit 1, will end in digit 1.

= **1**, = 133**1**

→ Cube of a number ending in digit 2, will end in digit 8.

=** 8**, = 172**8**

→ Cube of a number ending in digit 3, will end in 7.

= 2**7**, = 219**7**

→ Cube of a number ending in digit 4, will end in digit 4.

= 6**4**, = 274**4**

→ Cube of a number ending in digit 5, will end in digit 5.

=12**5**, = 337**5**

→ Cube of a number ending in digit 6, will end in digit 6.

= 21**6**, = 409**6**

→ Cube of a number ending in digit 7, will end in digit 3.

= 34**3**, = 491**3**

→ Cube of a number ending in digit 8, will end in digit 2.

= 51**2**, = 583**2**

→ Cube of a number ending in digit 9, will end in digit 9.

= 72**9**, = 685**9**

→ Cube of a number ending in digit 0, will end in digit 0.

= 100**0**, = 800**0**

*☆☆ From above we can observe that the Cube of the numbers ending in digits 0, 1, 4, 5, 6 and 9 are the numbers ending in same digits.*

→ Cube numbers can only have odd number of zeros at the end which must be triple the number of zeros at the end of the number whose cube it is.

=1**000**, = 8**000, ** =27**000**, = 1**000000000**

## Cube Roots

↪ Finding cube roots are the inverse operation of cubing.

*Cube root of a perfect cube number is the number whose perfect cube the number is.*

8 = 2×2×2, so we say that cube root of 8 is 2.

27 = 3×3×3, so we say that cube root of 27 is 3.

→ The symbol denotes the cube root.

Statement |
Inference |
Statement |
Inference |

## Finding Cube roots

#### Prime Factorisation method

→ Write the given number as product of its prime factors

→ Group the prime factors in triples.

→ Express the tripled prime factors in exponential form of power of 3

→ Take the power of 3 as common

→ Put before the number in LHS and remove the power in RHS to get the required value.

e.g. 216 = 2×2×2×3×3×3

⇒ 216 = __2×2×2__×__3×3×3__

## Problems with Solution

• Problems on Cube Numbers

– Finding Cube of a Number

– Finding the least Number to multiply / divide from a number to obtain a perfect cube.

– Word Problems.

• Problems on Cube Roots

– Finding Cube Roots by Factorisation Method

– Estimating Cube Roots by grouping digits of Numbers.