Square Roots- Calculation Methods and Examples
What Is a Square Root?
A square root is the number that, when multiplied by itself, gives you the original number. If √9 = 3, it's because 3 × 3 = 9. That's it. Nothing fancy.
Every positive number has two square roots — one positive, one negative. √16 = 4 and also -4. The positive one is called the principal square root. Most calculators and math problems default to this one.
Square Root Symbols and Notation
The radical symbol (√) tells you to find the square root. The number inside is called the radicand. If there's a small number (index) outside the radical, that tells you which root to find — but for square roots, there's no number written because it's assumed to be 2.
- √25 = 5 (principal square root)
- -√25 = -5 (negative square root)
- ±√25 = ±5 (both square roots)
How to Calculate Square Roots: 5 Methods
You have options here. Some are faster, some are more educational, and one is just for when you're stuck with pen and paper.
Method Comparison
| Method | Best For | Speed | Accuracy |
|---|---|---|---|
| Calculator | Any number, quick answers | Instant | Perfect |
| Prime Factorization | Perfect squares, small numbers | Fast for small radicands | Exact |
| Long Division Method | Manual calculation, learning | Slow | Exact |
| Estimation | Approximate answers, large numbers | Very fast | Approximate |
| Newton's Method | Programmatic/iterative solutions | Fast with iterations | Very accurate |
Getting Started: Calculate Square Roots Step by Step
Using a Calculator
This is what 99% of people do. Punch in the number, hit the √ button, done. Make sure you're using the right button — some calculators have separate buttons for square root versus other roots.
Finding Square Roots by Prime Factorization
This works when the number is a perfect square. Break the number into its prime factors, then pair them up. Each pair comes out of the radical as one number.
Example: √144
- 144 = 2 × 72
- 144 = 2 × 2 × 72
- 144 = 2 × 2 × 2 × 36
- 144 = 2 × 2 × 2 × 2 × 18
- 144 = 2 × 2 × 2 × 2 × 2 × 9
- 144 = 2⁴ × 3²
Pair the 2s: 2 × 2 = 4. Pair the 3s: 3 × 3 = 9. √144 = 4 × 3 = 12
Manual Estimation Method
When you can't use a calculator and the number isn't a perfect square, estimate.
Example: √50
- 7² = 49 (close)
- 8² = 64 (too high)
- √50 ≈ 7.07 (between 7 and 8, closer to 7)
For better precision: √50 ≈ 7.0710678...
The Long Division Method
This is tedious but accurate. Here's the process without going full classroom mode:
- Group digits in pairs from the right
- Find the largest number whose square fits the first group
- Subtract and bring down the next pair
- Double your current result and figure out what digit fills the next spot
- Repeat until satisfied
It works. It's just slow. Most people don't bother with this unless they're learning or stuck on a desert island without a calculator.
Square Roots of Non-Perfect Squares
Most numbers don't have clean square roots. √2, √3, √5, √7 — these go on forever with no repeating pattern.
- √2 ≈ 1.41421356...
- √3 ≈ 1.73205081...
- √5 ≈ 2.23606798...
- √7 ≈ 2.64575131...
These are called irrational numbers. The decimal never terminates and never repeats. You don't need to memorize them — just know they exist and use approximations when required.
Common Square Root Values to Know
| Number | Square Root | Number | Square Root |
|---|---|---|---|
| 1 | 1 | 49 | 7 |
| 4 | 2 | 64 | 8 |
| 9 | 3 | 81 | 9 |
| 16 | 4 | 100 | 10 |
| 25 | 5 | 121 | 11 |
| 36 | 6 | 144 | 12 |
Practical Uses of Square Roots
Square roots show up in real situations more than you'd expect:
- Geometry — Finding the diagonal of a rectangle or square
- Statistics — Standard deviation involves square roots
- Physics — Kinetic energy, wave calculations, distance formulas
- Engineering — Structural calculations, signal processing
- Finance — Volatility calculations, some interest formulas
Quick Rules and Properties
- √(a × b) = √a × √b — you can split the radicand
- √(a ÷ b) = √a ÷ √b — same for division
- √a² = |a| — the absolute value of a
- √a + √b ≠ √(a + b) — don't combine under one radical
- √0 = 0 — zero is its own square root
- √negative numbers = imaginary numbers (i√positive)
Solving Square Root Equations
When you see √x = something, square both sides to isolate x.
Example: √x = 5
Squaring both sides: (√x)² = 5² → x = 25
Always check your answer. Plug it back in: √25 = 5 ✓
Watch out for extraneous solutions — if you square an equation, you might introduce answers that don't actually work. Test everything.
Square Roots in Programming
Most programming languages have a built-in square root function:
- Python: math.sqrt(144) → 12.0
- JavaScript: Math.sqrt(144) → 12
- Excel: =SQRT(144) → 12
- C++: sqrt(144) → 12.0
What About Cube Roots and Beyond?
The same logic applies. A cube root (∛) is the number that, multiplied by itself three times, gives you the original. ∛8 = 2 because 2 × 2 × 2 = 8. Fourth roots work the same way — √[4]16 = 2 because 2⁴ = 16.
The calculation methods scale, but square roots are what you'll encounter 90% of the time.
The Bottom Line
Square roots aren't complicated. The concept is simple — find what number multiplied by itself gives you X. The execution is where it varies. Use a calculator for speed. Know prime factorization for perfect squares. Understand estimation for everything else.
That's it. No more to it.