## Arithmetic of square and square roots

### Square : raised to power 2

The result of multiplication of a number with itself is called a square. The square of
• 2 is 4 (2×2)
• 3 is 9 (3×3)
• 4 is 16 (4×4)
• 5 is 25 (5×5)
• 2.5 is 6.25 (2.5×2.5)
• 1.2 is 1.44

### Perfect Square : made with one number

When the square of a natural number is a natural number it is called a perfect square. The perfect square of
• 2 is 4 (2×2)
• 3 is 9 (3×3)
• 4 is 16 (4×4)
• 5 is 25 (5×5)
• So, 4, 9, 16, 25 etc are perfect squares.
• A given number is a perfect square, if it can be expressed as the product of exact number of pairs of equal factors.
• To check whether 225 is a perfect square.
1. Find the prime factors of 225.
225 = 3×3×5×5.
2. Make pairs of them
225 = (3×3)×(5×5).
3. As, 225 can be expressed as exact pairs of equal factors.
∴ 225 is a perfect square.

### Properties of number's square

• The square of an even number is always an even number.
1. An even number can be represented as (2n+2).
Square of it is
(2n+2)2 = 4n2 + 8n + 4
= 2(2n2 + 4n + 2) an even number.
2. Square of 2 is 4, 4 is 16, 6 is 36. All are even numbers.
• The square of an odd number is always an odd number.
1. An odd number can be represented as (2n+1).
Square of it is
(2n+1)2 = 4n2 + 4n + 1
= 2(2n2 + 2n) + 1 an odd number.
2. Square of 3 is 9, 5 is 25, 7 is 49. All are odd numbers.
• The square of a proper fraction is less than the proper fraction.
(1/3) has square as (1/9) and (1/9)<(1/3)
The same thing holds when the proper fraction is expressed as decimal.

### Square Root of numbers

The square root of a number x is the number whose square is x. We denote the square root of a number x as √x. The square root of
• 4 is 2 |(2×2)
• 9 is 3 |(3×3)
• 16 is 4 |(4×4)
• 25 is 5 |(5×5)
• 6.25 is 2.5
• 1.44 is 1.2

### Factorization method : Square Root

The square root of a perfect square is found by this method. Let us take example of 900.
• Step 1: Resolve the given number into prime factors.
900 = 2 × 3 × 5 ×2 × 3 × 5.
• Step 2: Make pairs of equal factors.
900 = (2 × 2) × (3 × 3) × (5 × 5).
• Step 3: Choose one number from each pair.
The numbers chosen are 2,3 and 5.
• Step 4: Find product of all of the numbers.
√900 = 2 × 3 × 5 = 30.

## Algebra of square and square roots

### Square: variable multiplied by variable

1. When we multiply a variable with itself it is called a square of the variable. The square of a is a2.

2. (a + b)2 = a2 + 2ab + b2

3. (a + b + c)2 = a2 + b2 + c2 + 2ab + 2ac + 2bc

4. We can write a two digit number as (10a + b).

The square of this number is (10a + b)2 = 100a2 + 10×2ab + b2.

It can be written as (a2)(2ab)(b2).

Suppose we want to find the square of 13 then 132 is (1)(6)(9) or 169.

and square of 25 is (4)(20)(25) = (4)(22)(5) =(6)(2)(5) = 625.

5. We can write a three digit number as (100a + 10b + c).

The square of this number is (100a + 10b + c)2 = 10000a2 + 1000(2ab) + 100(b2+2ac) + 10(2bc) + c2.

It can be written as (a2)(2ab)(b2+2ac)(2bc)(c2).

Suppose we want to find the square of 133 then 1332 is
(1)(6)(15)(18)(9)
or (1)(6)(16)(8)(9)
or (1)(7)(6)(8)(9)
or 17689.

and square of 255 is (4)(20)(45)(50)(25)
= (4)(20)(45)(52)(5)
= (4)(20)(50)(2)(5)
= (4)(25)(0)(2)(5)
= (6)(5)(0)(2)(5)
= 65025.

6. We can write a three digit number with decimal point before the last digit as (10a + b + 10-1c).

The square of this number is (10a + b + 10-1c)2 = 100a2 + 10(2ab) + (b2+2ac) + 10-1(2bc) + 10-2c2.

It can be written as (a2)(2ab)(b2+2ac).(2bc)(c2).

Suppose we want to find the square of 13.3 then 13.32 is
(1)(6)(15).(18)(9)
or (1)(6)(16).(8)(9)
or (1)(7)(6).(8)(9)
or 176.89

and square of 2.55 is (4).(20)(45)(50)(25)
= (4).(20)(45)(52)(5)
= (4).(20)(50)(2)(5)
= (4).(25)(0)(2)(5)
= (6).(5)(0)(2)(5)
= 6.5025

### Square Root

1. The number which when multiplied by itself gives the square, the number is called the square root of the number. The square root of a2 is a.

2. The square root of a2 + 2ab + b2 is (a+b).

3. The square root of a2 + b2 + c2 + 2ab + 2ac + 2bc is (a + b + c).

4. We can represent a two digit number as (10a + b)or(a)(b). The square of it is (100a2 + 20ab + b2) or (a2)(2ab)(b2). The figure describes how to find the square root of a number whose square root is a two digit number. For method see below in arithmetic.

5. We can represent a two digit number as (100a + 10b + c) or (a)(b)(c). The square of it is 10000a2 + 1000(2ab) + 100(b2+2ac) + 10(2bc) + c2 or (a2)(2ab)(b2+2ac)(2bc)(c2). The figure describes how to find the square root of a number whose square root is a three digit number. For method see below in arithmetic.

## Arithmetic

1. Looking at the above two procedures to find the square root of two and three digit numbers. We can follow the following procedure to find the square root of any number. This method is called the division method.

1. Group the digits from right into pairs. If the number has a decimal part, make pairs to both sides of decimal. If the decimal part has a digit left, add a zero to it.

2. Find  a number whose  square is less than or equal to the first pair or the remaining digits after forming pairs. take it as divisor and quotient.

3. Subtract the product of divisor and quotient from the first pair or the remaining digits after forming pairs.

4. Bring down the next pair to the right of the remainder. This is the new dividend.

5. Take the twice of quotient below on the left of the new dividend.

6. The new divisor is obtained by annexing the twice of the quotient by a digit. The digit is such that the product of new divisor and this digit is less than or equal to the new dividend.

7. Annex the new digit to the top quotient.

8. Subtract the number obtained by multiplying.

9. Repeat the process.

10. In case of taking quotient to decimal, add zeros to right of remainder in pairs.

11. Square root of 17689 is 133.

12. Square root of 1.7689

13. Square root of 176.89

14. Square root of 2 is 1.414...

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