Finding the square root of multidigit numbers can be done in several ways, depending on your preferences and tools available. Here are some common methods:

1. Prime Factorization Method (for perfect squares):

• Break the number into its prime factors.
• Pair up identical prime factors.
• Take one factor from each pair and multiply them together.
• If no unpaired factors remain, the result is the square root. If unpaired factors exist, the number is not a perfect square.

Example: √144
Prime factors of 144 = $2^4 \times 3^2$.
Pairing gives $(2 \times 2)$ and $(3 \times 3)$.
Square root = $2 \times 3 = 12$.

---

2. Long Division Method (useful for non-perfect squares):

• Group the digits of the number in pairs from right to left (or left to right for decimals).
• Find the largest integer whose square is less than or equal to the first group.
• Subtract the square of the chosen integer from the first group.
• Bring down the next pair of digits, and double the current divisor.
• Find a new digit to append to the divisor to form a new number that divides the current dividend as closely as possible.
• Repeat until the desired precision is achieved.

Example: √529

• Group digits: $5 | 29$.
• Largest integer ≤ √5 is 2. Subtract $2^2 = 4$.
• Bring down 29 → new dividend: 129.
• Double divisor: 2 × 2 = 4.
• New divisor digit: 3 (43 × 3 = 129).
• Result: 23.

---

3. Using a Calculator:

• Enter the number and press the square root (√) function to get the result instantly.
• This is the fastest and most practical method for large numbers.

---

4. Approximation and Iterative Methods:

• Newton-Raphson Method: Start with an initial guess $x_0$, then iterate using the formula:
  $x_{n+1} = \frac{1}{2} \left( x_n + \frac{N}{x_n} \right)$
  until the difference between successive approximations is small.
• Example: Find √50. Start with $x_0 = 7$:
  $x_1 = \frac{1}{2} \left( 7 + \frac{50}{7} \right) = 7.071$.
  Repeat for higher precision.

---

5. Estimation Method:

• Find the two perfect squares the number lies between.
• Approximate based on proximity to those squares.

Example: √200 lies between $√196 = 14$ and $√225 = 15$.
Approximation: ~14.1.

Would you like details or examples of any specific method? Here are some relative questions to expand your understanding:

1. What is the advantage of using the long division method over a calculator?
2. How does the Newton-Raphson method ensure precision?
3. How can you quickly identify if a number is a perfect square?
4. Can you explain how to apply the prime factorization method with decimals?
5. What are the practical uses of estimating square roots?

Tip: For mental math, memorizing square roots of common perfect squares (e.g., 4, 9, 16, 25, etc.) can greatly speed up approximations.