Squares and Square Roots

🔹 1. What is a Square?

When a number is multiplied by itself, the product is called a square number or perfect square.
👉 Example:
- ( 2² = 4 )
- ( 3² = 9 )
- ( 5² = 25 )

✅ General Form:
If ( a ) is any number, then
a² = a X a

🔹 2. Properties of Square Numbers

The square of an even number is even.
- Example: ( (4)² = 16 )
The square of an odd number is odd.
- Example: ( (3)² = 9 )
The number of zeros at the end of a perfect square is always even.
- Example: ( (20)² = 400 ) → 2 zeros
The square of a number ending in 2, 3, 7, or 8 never ends with 2, 3, 7, or 8.
The square of a number ending in 1, 4, 5, 6, or 9 always ends in 1, 4, 5, 6, or 9 respectively.
Squares of natural numbers are always positive.
Sum of first ( n ) odd numbers = ( n² )
1 + 3 + 5 + 7 +………….+ (2n – 1) = n²
👉 Example: ( 1 + 3 + 5 + 7 + 9 = 25 = 5^2 )

🔹 3. Perfect Squares between 1 and 100

1	1
2	4
3	9
4	16
5	25
6	36
7	49
8	64
9	81
10	100

👉 Total perfect squares from 1–100 = 10

🔹 4. Patterns Related to Squares

( 1² = 1 )
( 2² – 1² = 3 )
( 3² – 2² = 5 )
( 4² – 3² = 7 )

✅ Difference of consecutive squares = consecutive odd numbers

🔹 5. What is a Square Root?

The square root of a number is the value which, when multiplied by itself, gives the original number.

sqrt{a²} = a

👉 Example:

√9 = 3
√49 = 7

✅ Symbol: √

🔹 6. Properties of Square Roots

1️⃣ √(a × b) = √a × √b

2️⃣ √(a ÷ b) = √a ÷ √b (b ≠ 0)

3️⃣ √(a²) = a

4️⃣ (√a)² = a

✅ A perfect square always has an integer square root.

🔹 7. Methods to Find Square Roots

🟢 (A) By Repeated Subtraction Method

The sum of the first n odd numbers = ( n² ).
Subtract consecutive odd numbers from the given number until you reach 0.
The number of subtractions = square root.

👉 Example: Find √49
49 − 1 = 48
48 − 3 = 45
45 − 5 = 40
40 − 7 = 33
33 − 9 = 24
24 − 11 = 13
13 − 13 = 0
✅ Number of steps = 7 → √49 = 7

🟢 (B) By Prime Factorization Method

Step 1: Find prime factors of the given number.
Step 2: Make pairs of equal factors.
Step 3: Take one factor from each pair.
Step 4: Multiply them together.

👉 Example: Find √144
144 = 2 × 2 × 2 × 2 × 3 × 3
= (2 × 2) × (2 × 2) × (3 × 3)
√144 = 2 × 2 × 3 = 12
✅ √144 = 12

🟢 (C) By Long Division Method

Used for large numbers that are not easy to factorize.

Steps:

Group digits in pairs from right to left (for whole number part).
Find the largest number whose square ≤ first group.
Subtract and bring down the next pair.
Double the quotient and find a suitable digit ‘x’ such that
(20×quotient + x) × x ≤ dividend
Repeat the process till required decimal places.

👉 Example: Find √529
Step 1: Group digits → (5)(29)
Step 2: 2² = 4 (≤5) → remainder 1
Step 3: Bring down 29 → 129
Step 4: Double quotient = 2×2 = 4 → Try 43×3 = 129
✅ √529 = 23