**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.