## Cube & Cube-Root of any number by easy & fast way

We are applying Vedic formula for finding Cube & Cube-Root of any number **by observation only**. First we study the method to find cube of any number & then study to find Cube-Root of perfect cube.

**Cube of number-**

Vedic sutra Anurupya is applicable to find the cube of any number. As usual we will explain the steps while solving the sum. Read it 2-3 time carefully to understand the method. once you understand the method you can solve the sums in few seconds.

**Ex. i) 42 ^{3 }**

^{ }**St**ep 1: Very first step is divide the number in to two parts by slash, 4/2

Here a = 4 & b = 2

**Step 2:** Now find the value of first Row. If we move from right to left-

1) Left hand digit of first row is 4^{3} = 64.

**Now we move from left to right with ratio b/a i.e 2/4 = ½**

2) 2^{nd} digit in first row is = (½) x first digit

= (½) X 64 = 32

3) 3^{rd} digit in first row is = (½ ) x Second digit

= (½) x 32 = 16

4) 4^{th} digit in first row is = (½) x Third digit

= (½) x 16 = 08 = 2^{3}

**Now our first row is, 64 32 16 8**

**Step 3: Second row is obtained by doubling the middle term of first row & add them**.

**Step 4: **Now to obtained final answer we put 3 Zero behind the first term from LHS, 2 zero behind second term, one zero behind third term & no zero behind last term then add them**. Adding zero rule is same for 2 & 3 digit number.**

**Answer, 42 ^{3} = 74088**

**Ex. ii) 23 ^{3}**

^{ }**Step 1: **Consider a =2 & b =3

**Step 2:**

1) Left most digit in first row is a^{3} = 2^{3} = 8

2) Second digit = (b/a) x first digit

= (3/2) x 8 = 12

3) Third digit = (3/2) X 12 =18

4) Fourth digit = (3/2) x 18 = 27 = b^{3}

**Now our first row is, 8 12 18 27**

**Step 3:** **Second row is obtained by doubling the middle term of first row & add them. **

**Step 4: Add appropriate zero on the final row & add them.**

** **8000 + 3600 + 540 + 27 = 12167

**Answer**,** 23 ^{3} = 12167**

**Similarly,**

**Ex.iii) 103 ^{3 }**= 10/3 (For 3 – digit number, LHS of slash contain 2-digit)

^{ }**Here a=10 & b = 3**

1) First digit of first row = 10^{3} =1000 & ratio is b/a = 3/10

2) Second digit = (b/a) x first digit

= (3/10) x 1000 = 300

3) Third digit = (3/10) X 300 =90

4) Fourth digit = (3/10) x 90 = 9 = b^{3}

**Now our first row is, 1000 300 90 9**

**Step 3:** **Second row is obtained by doubling the middle term of first row & add them.**

**Now to obtained final answer add the total row by adding zero on terms,**

1000000 + 90000 + 2700 + 9 =1092727

**Answer,** **103 ^{3} = 1092727**

**Cube root of perfect cube-**

We will consider perfect cube only to find the cube root as practically it asked more. Initially we memorized the cube of number 1 to 10.

Now by observing the last digit of above table we make one more table

From above table we can conclude that cube of number 0,1,4,5,6 & 9 end with same number, where as cube of 2 end with 8 & cube of 3 end with 7 & vice versa.

**Ex. i) Find the cube root of 85184 **

**Step 1: **Put a slash before last 3 digit

** 85/184**

**Step 2: **Here last digit is 4, & we know from table 2, last digit of cube is 4, So last digit of cube root will be also 4.

So, we get our last digit _4.

**Step 3:** To find first digit of the answer, we observe LHS of the given cube. It is 85 here. 85 lies between –

**Step 4: **Out of these two number 4 & 5 we take smaller number as first digit of our answer.** (Always take smaller number from two)**

** **So, our first digit is 4.

** Answer, cube root of 85184 = 44.**

**Ex. ii) Find the cube root of 1728**

** Step 1: **1/728

** Step 2: **Last digit of answer = _2 (From table 2)** **

** Step 3: **LHS part of slash is 1, which is cube root of 1

** Step 4: **So, our first digit = 1.

** Answer, cube root of 1728 = 12. **

** **

**Ex. iii) Find the cube root of 681472.**

**Step 1: **681/472

**Step 2: **Last digit of Answer = _8 (From table 2)** **

**Step 3: **681 lies between 512 & 729, which are the cube of 8 &9

**Step 4: From 8 & 9 smaller number is 8, So our first digit is 8.**

**Answer, cube root of 681472 = 88 **

** **

**Ex. iv) Ex. iii) Find the cube root of 1225043**

**Step 1: **1225/043

**Step 2: **Last digit of Answer = _7 (From table 2)** **

**Step 3: **1225 lies between 1000 & 1331, which are the cube of 10 & 11.

**Step 4: **From 10 & 11 smaller number is 10, So our first digit is 10.

**Answer, cube root of 1225043 = 107.**

** **

**Find this Link to Study more:**

1.Find the square & Square-root of any number by observation only

2.Base Method of Multiplication: Multiply a number near the base in 3-4 seconds

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