**Square**

The square of a number is the result of multiplying a number by itself. For example the square of 6 is 6×6.

Squaring is raising to the power 2, it is denoted by a superscript 2. E.g 6², 4²

A square of a number is also called a perfect square. In algebra, it is often generalized to polynomials. For example, the square of the linear polynomial x + 1 is the quadratic polynomial (x+1)² = x²+ 2x + 1².

**Square ****Roo**t

The square root is just the opposite of the square. It is equal to a number, which when squared gives the original number or a number which on multiplication by itself gives the original number.

For example, the square of 4 is 16, 4 × 4 = 16 and the square root of 16, √16 = 4.

It is easy to find the square root of perfect numbers, e.g. 16 is a perfect number so the square root of 16 is 4, but to find the square root of an imperfect number like 3, 7, 5, etc., it needs some tricks to find the root.

Vedic maths is here to help you to find squares or square roots in easy and effective ways.

**Square Root Rules To Remember**

**Square**

**Square of 11, 111, 1111….**

Squaring 11 or all multiples if 11 is very easy as compared to others. Understand carefully there are two ones to make 11. So write upto two numbers and return back.

- ( 11 )² = 1 2 1

- ( 111 )² = 1 2 3 2 1 ( here 3 one’s are present so write upto 3 numbers, means 123 and then come back after 3 that is 2 and 1)

- ( 1111 )² = 1234321

**Square of 22, 33, 44 ……99, 222, 333…..999 etc.**

Here you can apply same formula as 11 after taking out actual number.

- (22)² = 2² × (11)² = 4 × 121 = 4 8 4

- (33)² = 3² × (11)² = 9 × 121 = 1089

- (77)² = 7² (11)² = 49 × 121 = 5929

- (99)² = 9² (11)² = 81 × 121 = 9801

- (222)² = 2² × (111)² = 4 × 12321 = 49284

- (444)² = 4² × 12321 = 16 × 12321 = 197136

**Square using (a+b)² = a² + 2ab + b²**

(57)² = 5² | 2 × 5 × 7 | 7²

= 2 5 **|** 7 0** | ** 4 9 – use Balancing rule

|_+_| |_+_|

= 3 2 4 9

(46)² = 4² | 2 × 4 × 6 | 6²

= 16 **|** 48 **|** 36

= 2116

(89)² = 8² + (2 × 8 × 9) + 9²

= 6 4 **|** 1 4 4 **|** 8 1

= 7 9 2 1

⬆️ ⬆️ ⬆️

⬆️ ⬆️ (8+4=12-one carry to 14)

(6+1=7)(15+4=19-one carry to 6)

**Square of 25, 35, 45 …….95, 125, 135 …**

**Step 1**: If unit digit is 5 then, square 5, it will give 25

**Step 2**: Use Ekadhikena Purvena for rest of the number, it means add 1 to the previous number and multiply each other.

**Step 3:** Join both the results together to get answer.

**Example 1:** (65)²

(65)² = (6 + 1) × 6 and (5 × 5)

= 7 × 6 and 25

= 4225 – Answer

**Example 2**: (95)²

(95)² = (9 + 1) × 9 and 25

= 10 × 9 and 25

= 9025 – Answer

**Example 3:** (135)²

(135)² = (13 + 1) × 13 and 25

= 14 × 13 and 25

= 18225

**Example 4:** (675)²

(675)² = (67 + 1) × 67 and 25

= 68 × 67 and 25

⬆️

(To solve fast use criss-cross multiplication method. * By using criss-cross method you can solve whole 675²)

= 455625

**Square using Base system**

Squaring the number which is near the base is easy to solve. It may be lesser or more. For example 98 is near 100 and it has 2 lesser which is also called complement. Similarly 106 is near the base 100 which has 6 more.

**Step 1: **Separate the excess or lesser numbers

**Step 2:** Add or subtract the excess or lesser number to actual number

**Step 3:** Multiply differences

**Step 4**: Join results obtained from step 1 and step 2

**Example 1:**

(97)² 97 —- (-3)

Subtract ↗️

97 —- (-3)

__________

97-3=94 09 》 Answer 9409

⬆️

(Putting 0 because we are taking 100 base which is having two zeros that means two digits, so here 09 are two digits)

**Example 2: **

(104)² =

104 —- 4

104 —- 4

__________

104+4=108 16 – Already two digits

》Answer is 10816

**Example 3: **

(993)² = 993 —- (-7)

993 —- (-7)

______________

993-7=986 049 》 Answer is 986049

⬆️

(Extra zero because of 1000 base)

**Example 4: **

(1008)² = 1008 —- +8

1008 —- +8

______________

1008+8=1016 064 》 Answer 1016064

**Square Root**

For solving square root in seconds, you need to remember or to be precise mugup squares from 1 – 20. Otherwise you have to write every time.

Squares of 1 – 15:

1² = 1

2² = 4

3² = 9

4² = 16

5² = 25

6² = 36

7² = 49

8² = 64

9² = 81

10² = 100

11² = 121

12² = 144

13² = 169

14² = 196

15² = 225

Now remember:

1 – is present in square of 9 and 1

4 —– 2 and 8

5 —- 5

6 —– 6 and 4

9 —– 7 and 3

Only these numbers come in squares of two numbers. Now let’s see how we can implement these numbers to solve

**Example 1:** √4489

√4489

**Step 1**: Make pairs, arranged in two-digit groups from right to left.

In this case make 44 and 89 as pairs.

**Step 2:** Only take the unit digit and see it fall under the square of which numbers. In this case the unit digit is 9 and it comes in squares of 3 and 7.

3

↗️

√44 89

↘️

7

**Step 3**: Now check pair 44 comes between which number square. In this case 44 comes under 36 (6²) and 49 (7²).

**Step 4**: Take the lower number. In this case 6 is lower than 7. So take 6, than make 63 and 67 (squaring of numbers 3 and 7 ends with 9)

**Step 5**: Last step is to check 44 is near to which number 36 (6²) or 49 (7²). In this case 44 is near 49. This means we take 67 (out of 63 and 67)

》 Answer is 67

**Example 2:** √6724

√ 67 24

**Step 1:** Pairs 67 and 24

**Step 2:** Take the last unit digit, which is 4. Square of numbers 2 and 8 ends with 4.

**Step 3:** Pair 67 comes between 64(8²) and 81(9²).

**Step 4:** Take lower numbers out of 8² and 9². Here it is 8. Make 82 and 88.

**Step 5**: Check Pair 67 is near to which number 64 or 81. Here 67 is near 64.

This means 82 is the answer

》 Answer is 82

**Frequently Asked Questions **

Question 1: **What is Square** ?

**Answer**: The square of a number is the result of multiplying a number by itself. For example the square of 6 is 6×6.

Squaring is raising to the power 2, it is denoted by a superscript 2. E.g 6², 4²

A square of a number is also called a perfect square. In algebra, it is often generalized to polynomials. For example, the square of the linear polynomial x + 1 is the quadratic polynomial (x+1)² = x²+ 2x + 1².

Question 2: What is Square Root?

**Answer:** The square root is just the opposite of the square. It is equal to a number, which when squared gives the original number or a number which on multiplication by itself gives the original number.

For example, the square of 4 is 16, 4 × 4 = 16 and the square root of 16, √16 = 4.

It is easy to find the square root of perfect numbers, e.g. 16 is a perfect number so the square root of 16 is 4, but to find the square root of an imperfect number like 3, 7, 5, etc., it needs some tricks to find the root.

**Final Words**

From the above article you must have learned very effective techniques of square and square root. Perhaps applying these techniques needs a lot of practice, so as to do any maths square or square root problems fast, within 2 to 5 seconds.