Cubes: Vedic Maths | Mathematics (Maths) Class 8 PDF Download

Cube

Cube is the product of a number multiplied by its square. When we multiply any number three times, the resultant number is called the cube of the original number.
In other words, when a number raised to exponent 3 is known as the cube of that number. It is represented by a superscript 3.
For example, the cube of 2 is 8 (2 × 2 × 2), it can be written as 2³. Similarly, the cube of 5 is 125 (5 × 5 × 5) and it can be written as 5³.

Perfect And Imperfect Cube

A perfect cube is a number or an integer which is equal to the number, multiplied by itself, three times. If x is a perfect cube of y, then we can write x = y3. So when we take out the cube root of a perfect cube, we get a natural number. Hence, 3√x = y.
For example, 27 is a perfect cube because 27 = 3 × 3 × 3. Whereas, 28 is not a perfect cube because there is no number, which, when multiplied three times gives the product 28.

Cube of Number

For finding a Cube of any number we need to use two Vedic maths sutras

• Anurupyena Sutra (Specific Technique)
• Yavadunam Sutra (Specific Technique)

Cube by applying Anurupyena Sutra
Algebraic expression
(x + y)³ = x³ + 3x²y + 3y²x + y³

Example 1: (23)³

Ans: (23)³ — Apply a³ + 3a²b + 3b²a + b³ formula
= 2³ + (3 × 2² × 3) + (3 × 3² × 2) + 3³
= 8 + 36 + 54 + 27
= 8 | 3 6 | 5 4 | 2 7 (use Balancing rule)
= 12 1 6 7

Example 2: (27)³

Ans: = 2³ + (3 × 2² × 7) + (3 × 7² × 2) + 7³
= 8 + 84 + 294 + 343
= 8 | 8 4 | 2 9 4 | 3 4 3
= 1 9 6 8 3

We can solve it another way
(15)³
Start from left side:
Step 1: First write the first digit as it is. 1
Step 2: Multiply one with five 1 × 5 = 5
Step 3: Multiply again five with five 5 × 5 = 25
Step 4: Again multiply 25 with 5. 25 × 5 = 125
Step 6: Write in series 1 5 25 125
Step 7: Multiply 5 × 2 = 10 and 25 × 2 =50
Step 8: Write 10 and 25 just below 5 and 25
Step 9: Add digits to get the answer

Cube of 12, 13, 15 — 19

Example 1: (15)³

Ans: = 1 5 25 125
× × 10 50 ×
___________________
1 | 15 | 75 | 125 – Balancing Rule
3 3 7 5 – Answer
(Starting from right side drop 5, carry 12 to 5 add 12 + 5 = 17, drop 7 add adjacent 7 + 1 = 8, carry 8 to next 5, add 8 + 5 = 13, drop 3 add remaining 1 with 1 gives 1 + 1= 2, carry 2 to next 1, add 2 + 1 = 3)

Example 2: (16)³

Ans: = 1 1 6 36 216
× × 32 72 ×
_______________________
1 | 18 | 108 | 216
4 0 9 6 – Answer

Cube of 22, 33, 44, 55 —— 99

Example 1: (22)³

Ans: In this case the same numbers start from the left, but this time write the cubes of each number. 2³ = 8
(22)³
= 8 8 8 8
× × 16 16 ×
_________________
8 | 24 | 24 | 8
10 6 4 8 – Answer

Example 2: (66)³

Ans: 216 216 216 216
××× 432 432 ×
_____________________________
216 | 648 | 648 | 216
28 7 4 9 6 – Answer

Cube of 21, 31, 41, 51 —– 91

In this case start from the right side. Other things remain same

Example 1: (21)³

Ans:
= 8 4 2 1
× × 8 4 ×
__________________
—8 | 12 | 6 | 1
9 2 8 1 – Answer

Example 2: (41)³

Ans: 64 16 4 1
×× 32 8 ×
___________________
64 | 48 | 12 | 1
6 8 9 2 1- Answer

Cube of any two digit number other then previous numbers 24, 32, 46 — 98

In this case start from the left side. E.g. (32)³
Step 1: Write cube of first number 3³ = 27
Step 2: Make the square of the first number and multiply it with the second number.

3² × 2 = 18
Step 3: Make the square of the second number and multiply with the first number.
2² × 3 = 12
Step 4: Write cube of second number 2³ = 8
Step 5: Write in series 27 18 12 8
Step 6: Multiply 18 and 12 with 2 and write the answers below. 18 × 2 = 36, 12 × 2 = 24
Step 7: Add numbers and apply the Balancing Rule.

Example 1: (32)³

Ans: 27 18 12 8
×× 36 24 ×
______________________
27 | 54 | 36 | 8
3 2 7 6 8 – Answer

Example 2: (62)³

Ans: 216 72 24 8
××× 144 48 ×
_____________________
216 | 216 | 72 | 8
23 8 3 2 8 – Answer

One more method we can use for a two digit number is first square the number and multiply again with the same number.

Example: (34)³

Ans: (34)³ = (34)² × 34
= (3² + 3 × 4 × 2 + 4²) × 34
= (9 + 24 + 16) × 34
= 9 | 24 | 16 × 34
= 1156 × 34- Apply Criss-Cross Method
= 39304

Note: This method you can apply for three digit numbers