Last updated on July 14th, 2024 at 05:55 pm
Cube and Cube root
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 Root
Cube root is the number that needs to be multiplied three times to get the original number. Cube root is an inverse operation of the cube of a number.
It is the reverse process of the cube of a number and is denoted by ∛. Here the word Root represents the primary source or origin of the cube. The other way to denote cube root is to write 1/3 as the exponent of a number.
So, we just need to find out the cube of which number should be taken to get the given number. The cube root formula is ∛x = y. Where y is the cube root of x.
We can solve cube or cube root of any number just by observation only by applying Vedic maths formula.
First we will learn the method to find a cube of any number & then we will learn how to find the Cube-Root of the perfect cube.
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
(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
》 Answer is 12167
Example 2: (27)³
= 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
= 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
= 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
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
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
= 8 4 2 1
× × 8 4 ×
__________________
—8 | 12 | 6 | 1
9 2 8 1 – Answer
Example 2: (41)³
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
27 18 12 8
×× 36 24 ×
______________________
27 | 54 | 36 | 8
3 2 7 6 8 – Answer
Example 2: (62)³
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
(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
This method you can apply for three digit numbers
Cube Root of Numbers
To calculate cube root we need to Memorize the cubes of unit digit and its relations from 1 to 10. In fact, cube root is easier than square root, provided you have a cube of 1 to 10 in your mind.
Remember the following cubes in mind
1³ = 1
2³ = 12
3³ = 27
4³ = 32
5³ = 125
6³ = 216
7³ = 343
8³ = 512
9³ = 729
10³ = 1000
Also remember:
When the last digit of a cube root is 8 then the unit digit will be 2, and vise versa
2 —— 8
8 —— 2
Same case goes with 7 and 3
7 —— 3
3 —— 7
Other than these 4 numbers 2, 3, 7, 8 all other digits give the same unit digit. E.g.
1 — 1
4 — 4
5 — 5
6 — 6
9 — 9
To Learn Vedic Maths Multiplication Click Here
Now let’s see how we can make use of these.
Example 1: ∛12167
Step 1: Separate the number in two parts three digits each. If two digits remain at last, then put one 0 before the number and make it a group of three digits.
∛ 12167 = 012 , 167
Step 2: Here the last digit is 7 then the unit digit will be 3 (as discussed earlier)
Step 3: Now see the remaining 12 comes under which digit cubes. In this case 12 comes between the cube of 2³ and 3³. So take the lower one 2
Step 4: Join results obtained from step 2 and step 3. That is 2 and 3. So answer is 23
Example 2: ∛970299
Step 1: Two groups 970 and 299. Last digit is 9 so unit digit will also comes 9
Step 2: Group 970 comes under 9³ and 10³. So take smaller one 9
Step 3: Join results obtained from step 1 and step 2. 99 is the Answer
Example 3: ∛551368
∛551368
Step 1: 551 and 368. Last digit is 8 so unit digit will be 2
Step 2: 551 comes under 8³ (512) and 9³ (729). So take smaller one that is 8
Step 3: Join result of step 2 and step 1, that is 82. Answer is 82
Example 4: ∛19683 + ∛2197
Step 1: ∛19683 = 27
Step 2: ∛2197 = 13
Step 3: Add results obtained from step one and two. 27 + 13 = 40
Frequently Asked Questions
Q 1: What is Cube
Answer: 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.
Q2: What is cube root?
Answer: Cube root is the number that needs to be multiplied three times to get the original number. Cube root is an inverse operation of the cube of a number.
Q3: What is the difference between cube and cube root?
Answer:
Cube is the product of a number multiplied by its square.
Cube root is the reverse process of the cube of a number and is denoted by ∛.
Final Words
From the above article you learn how to find out the cube and cube root of any number. The only thing required here is regular practice which will make you master of Vedic maths cube and cube root. All the best.