You already know that $60^2=3600$. Using the pattern $(n+1{)}^2=n^2+2n+1$, write down $61^2$ without multiplying.

Use this for larger numbers (or non-perfect squares).

**Step 1**: Pair the digits from the right (e.g. 1849 → 18 49).
**Step 2**: Find the largest integer whose square ≤ first pair. Write it as the first quotient digit. Subtract its square.
**Step 3**: Bring down the next pair. Double the current quotient — that's your divisor's prefix. Find a digit $d$ such that (prefix × $10+d)\times d$ ≤ current dividend.
**Step 4**: Append $d$ to quotient, subtract, repeat.

Example: $\sqrt{1849}=43.(4^2=16$ ≤ 18; remainder 2; bring down 49 → 249; double 4 = 8; find $d$ where $(80+d)d$ ≤ 249; $d=3$ gives $83\times 3=249$.) Quotient = 43.