Cube:
It is a three-dimensional figure made of six equal square sides.

Cube number or Perfect cubes:
It is a number which is the product of three same numbers.
Example: Cube number of 2 will be 2 x 2 x 2 = 8. Thus, 8 is a cube number.

Cubes of some natural numbers:

Properties of Cube Numbers:
1. The cube of an even number will always be an even number.
Example: 8³ = 512, 12³ = 1728, etc.

2. The cube of odd number will always be an odd number.
Example: 7³ = 343, 19³ = 6589, etc.

3. If the cube number have x at its one’s digit or unit’s place then it always end with the digit as shown in the table below:

Example 1: Find the one’s digit for 27.
Solution: As the last digit of given number is 7, So the one’s digit for 27’s cube number will be 3.

Example 2: Find the one’s digit for 149.
Solution: As the last digit of given number is 9, So the one’s digit for 149’s cube number will be 9.

Interesting patterns of Cube Number:
1. Addition of consecutive odd numbers will give Cube Number-
1³         =          1          =          1
2³         =          8          =          3 + 5
3³         =          27        =          7 + 9 + 11
4³         =          64        =          13 + 15 + 17 + 19
5³         =          125      =          21 + 23 + 25 + 27 + 29

2. Cubes and their prime factors-
The prime factors of any cube number will be in pair of 3.
Example:
(i) 4³ = 64 = 2 x 2 x 2 x 2 x 2 x 2 = 2³ x 2³
(ii)12³ = 1728 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 = 2³ x 2³ x 3³

Note:
(i) If any prime factor is not in pair of 3, then it will not be a perfect cube.
(ii) Numbers like 1729, 4104, 13832, are known as Hardy – Ramanujan Numbers. They can be expressed as sum of two cubes in two different ways.

Example 1: Is 128 a perfect cube number?
Solution: After finding prime factors of 128, we can write 128 = 2 x 2 x 2 x 2 x 2 x 2 x 2
We can see that each prime factor is not in pair of 3. Hence, 128 is not a perfect cube number.

Example 2: Find the smallest number by which 675 must be multiplied to obtain a perfect cube.
Solution: On finding prime factors of 675, we have 675 = 3 x 3 x 3 x 5 x 5.
We can see that, triplets of 5 is missing. Hence, on multiplying given number by 5 we can have a perfect cube number. Thus, 675 x 5 = 3375 which is a perfect cube number.

Example 3: Find the smallest number by which 192 must be divided to obtain a perfect cube.
Solution: On finding prime factors of 192, we have 192 = 2 x 2 x 2 x 2 x 2 x 2 x 3.
We can see that, triplets of 3 is missing while other numbers have triplets. Hence, on dividing given number by 3 we can have a perfect cube number.
Thus, 192 / 3 = 64 which is a perfect cube number.

Cube Roots:
It is the inverse operation of finding a cube. Symbol ∛ represents a cube-root.
Example: ∛8 = 2, ∛216 = 6, etc.

Methods to find a cube root:
1. Prime factorisation method:
Follow the steps given below to understand this method:
Step 1: Find all the prime factors of given cube number.
Step 2: Make as many group of 3 for all common digit.
Step 3: Replace group of 3 by respective single digit.
Step 4: Product of these single digits will give the cube root.

Example 1: Find cube root of 3375.
Solution:
Step 1: Find all the prime factors of given cube number.
The prime factors of 3375 will be 5, 5, 3, 3, 3, 5.
Step 2: Making group of 3 for every common digit.
Here, we get (3 x 3 x 3) and (5 x 5 x 5)
Step 3: Replacing group of 3 by respective digit.
Thus, we get 3 and 5.
Step 4: Taking product of digits.
We get, 3 x 5 = 15.
Thus, ∛3375 = 15.

Example 2: Find cube root of 46656.
Solution: The prime factors of 46656 are 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3.
On grouping, we have 46656 = 2 x 2 x 2 x 2 x 2 x 2 x 3 x 3 x 3 x 3 x 3 x 3.
So, ∛46656 = 2 x 2 x 3 x 3 = 36.

2. Estimation Method:
Follow the steps given below to understand this method:
Step 1: For given cube number start making groups of three digits starting from the right most digit of the number. We can estimate the cube root of a given cube number through a step by step process.
Step 2: First group will give you the one’s (or unit’s) digit of the required cube root.
Step 3: Now take another group, let it be xyz. Find, a³ < xyz < b³. We take the one’s place, of the smaller number a³ as the ten’s place of the required cube root.
Step 4: The digits obtained in step 2 and step 3 will give the final result.

Example 1: Find the cube root of 17576.
Solution:
Step 1: We will form groups of three starting from the rightmost side of 17567.
Thus, the two groups formed will be 576 and 17.
Step 2: For the group 576, the number at unit’s place will be 6.
Step 3: The other group is 17. The group 17 lies between 2³ and 3³. Now, as per the method we will take the smaller number which is 2 in this case.
Step 4: Combining the two digits obtained in step 2 and 3, we get 26.
Thus, ∛17576 = 26.

Example 2: Guess the cube root of 4913.
Solution:The two groups of 4913 will be 4 and 913.
For group 913, as the last digit is 3 so its cube root will have 7 at its unit place.
For group 4, it will lie between 1³ < 4 < 2³. Taking the smaller number which is 1 in this case.
Thus, ∛4913 = 17.