Answer: Use one of several methods — recognize perfect squares or use prime factorization for exact integers, the long-division (manual square-root) method for digit-by-digit results, or Newton’s (Babylonian) method for fast decimal approximations. Below I explain each with examples and tips.
Quick overview
- If the number is a perfect square (1,4,9,16,25,…), its square root is the integer whose square gives the number.
- For integers, prime-factorization can give an exact integer/root-simplified form.
- For non-perfect or decimal numbers, use Newton’s method (fast convergence) or the long-division method (digit-by-digit).
- Calculators and programming languages provide built-in sqrt functions.
1) Perfect squares and prime factorization
Example: Find $\sqrt{3600}$.
- Prime factorize $3600 = 2^4 \times 3^2 \times 5^2$.
- Take one factor of each pair: $\sqrt{3600} = 2^2 \times 3 \times 5 = 4 \times 3 \times 5 = 60$.
Rule: Pair up identical prime factors; each pair contributes one factor to the square root.
2) Long-division (manual digit) method
Use this when you want exact decimal digits without a calculator.
Example: $\sqrt{1521}$.
- Group digits from the decimal point in pairs: 15 | 21.
- Largest square ≤ 15 is $3^2=9$. First digit 3. Subtract 15−9=6. Bring down 21 → 621.
- Double current root (3 → 6). Find digit $d$ so that $(60 + d)d \le 621$. Try $d=9$: $(60+9)9=69\times9=621$. So $d=9$.
- Root is 39 and remainder 0. So $\sqrt{1521}=39$.
(The method works similarly for decimals by adding pairs of zeros after the decimal point.)
3) Newton’s (Babylonian) method — fast for decimals
To approximate $\sqrt{S}$, iterate
$$x_{n+1}=\frac{x_n + S/x_n}{2}.$$
Example: approximate $\sqrt{20}$.
- Choose initial $x_0=5$ (since $5^2=25$).
- $x_1=(5 + 20/5)/2=(5+4)/2=4.5$.
- $x_2=(4.5 + 20/4.5)/2\approx(4.5+4.444444)/2\approx4.472222$.
- $x_3\approx4.472136$ → converges quickly to $\sqrt{20}\approx4.472136$.
Newton’s method typically doubles correct digits each iteration once close.
4) Estimation + refinement
- Find nearest perfect squares. E.g., $16<20<25$, so $\sqrt{20}$ is between 4 and 5.
- Interpolate or use one Newton step from an estimate to refine.
5) Using a calculator or programming
- In calculators use sqrt or x^(1/2).
- In many languages: Python:
import math; math.sqrt(20)or20**0.5.
6) Special notes
- For negative numbers, square roots are complex: $\sqrt{-a}=i\sqrt{a}$ for $a>0$.
- Property: $\sqrt{ab}=\sqrt{a}\sqrt{b}$ for nonnegative $a,b$; $\sqrt{a^2}=|a|$.
If you want, tell me a specific number (integer or decimal) and I’ll show one of these methods step-by-step for that number.