## Cubes and Cube Roots

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.

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# 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, = 1331

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

= 8, = 1728

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

= 27, = 2197

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

= 64, = 2744

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

=125, = 3375

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

= 216, = 4096

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

= 343, = 4913

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

= 512, = 5832

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

= 729, = 6859

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

= 1000, = 8000

☆☆ 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.

=1000, = 8000, =27000, = 1000000000

## 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.