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 2^{3}. Similarly, the cube of 5 is 125 (5 × 5 × 5) and it can be written as 5^{3}.
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.
For finding a Cube of any number we need to use two Vedic maths sutras
Cube by applying Anurupyena Sutra
Algebraic expression
(x + y)^{3} = x^{3} + 3x^{2}y + 3y^{2}x + y^{3}
Example 1: (23)^{3}
Ans: (23)^{3} — Apply a^{3} + 3a^{2}b + 3b^{2}a + b^{3} formula
= 2^{3} + (3 × 2^{2} × 3) + (3 × 3^{2} × 2) + 3^{3}
= 8 + 36 + 54 + 27
= 8  3 6  5 4  2 7 (use Balancing rule)
= 12 1 6 7
Example 2: (27)^{3}
Ans: = 2^{3} + (3 × 2^{2} × 7) + (3 × 7^{2} × 2) + 7^{3}
= 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
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
Example 1: (22)^{3}
Ans: In this case the same numbers start from the left, but this time write the cubes of each number. 2^{3} = 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
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
In this case start from the left side. E.g. (32)^{3}
Step 1: Write cube of first number 3^{3} = 27
Step 2: Make the square of the first number and multiply it with the second number.
3^{2} × 2 = 18
Step 3: Make the square of the second number and multiply with the first number.
2^{2} × 3 = 12
Step 4: Write cube of second number 2^{3} = 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
Example: (34)^{3}
Ans: (34)^{3} = (34)^{2} × 34
= (3^{2} + 3 × 4 × 2 + 4^{2}) × 34
= (9 + 24 + 16) × 34
= 9  24  16 × 34
= 1156 × 34 Apply CrissCross Method
= 39304
Note: This method you can apply for three digit numbers
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