1. Prime Factorization Method

• Express the number as a product of its prime factors.
• Pair the same prime numbers.
• Take one number from each pair and multiply them.

Example:
Find √144 using prime factorization.
144 = 2 × 2 × 2 × 2 × 3 × 3
√144 = 2 × 2 × 3 = 12

2. Long Division Method

Used when the number is not a small perfect square.

Step 1: Pairing the Digits

• Start grouping the digits of the number in pairs from right to left.
• Each pair (or single digit in the case of an odd number of digits) will be processed in steps.

Step 2: Find the Largest Square

• Identify the largest number whose square is less than or equal to the first group (or single digit).
• Write this number as the quotient and the divisor.
• Subtract its square from the first group and bring down the next pair of digits.

Step 3: Double the Quotient

• Take the quotient, double it, and write it as the new divisor (without the last digit).
• Find a digit that, when added to this divisor and multiplied by itself, is the largest possible without exceeding the number.

Step 4: Repeat the Process

• Continue the process until all digits have been used.
• If a decimal is needed, continue by adding pairs of zeros to the remainder and repeating the process.

3. Estimation Method

• Identify two perfect squares between which the number lies.
• Estimate the value between those two numbers.

Example:
Find √40.

• 36 < 40 < 49
• √36 = 6 and √49 = 7
• Since 40 is closer to 36, √40 ≈ 6.3

4. Using Repeated Subtraction

For small perfect squares, subtract successive odd numbers until you reach zero. The number of steps taken is the square root.

Example: Find √16.
16 – 1 = 15
15 – 3 = 12
12 – 5 = 7
7 – 7 = 0

Since 4 steps were taken, √16 = 4.

These methods provide different approaches to finding square roots efficiently based on the type of number.

Conceptual Understanding of Square Roots of Decimals

Definition:
The square root of a number is a number such that:

y² =x

Methods to Find the Square Root of a Decimal:

1. Convert to a Fraction:

   • Express the decimal as a fraction and find the square root of both numerator and denominator.

2. Estimation Method:

   • Identify two nearby perfect squares.

3. Long Division Method (For Precise Calculation):

   • A step-by-step approach similar to long division, used for non-perfect squares.

This approach ensures accurate calculations of square roots for decimal numbers.