Square Root Explained- Calculation Methods and Examples

What Is a Square Root, Exactly?

A square root is the number you multiply by itself to get another number. If √9 = 3, it's because 3 × 3 = 9. That's the whole concept. Nothing fancy.

Every positive number has two square roots—one positive, one negative. √9 gives you 3, but -3 squared also equals 9. Most calculators just show the positive version because that's what people usually want.

Zero is the only number with a single square root. √0 = 0.

Negative numbers don't have real square roots. √-9 isn't a real number. You'll see it written as 3i in math classes—that "i" means imaginary. Most everyday math ignores this.

How to Calculate Square Roots

You have four real options. Each works, depending on what you're doing.

1. Mental Math (Perfect Squares)

Memorize the basics. You'll use these constantly.

• 1² = 1, so √1 = 1
• 2² = 4, so √4 = 2
• 3² = 9, so √9 = 3
• 4² = 16, so √16 = 4
• 5² = 25, so √25 = 5
• 6² = 36, so √36 = 6
• 7² = 49, so √49 = 7
• 8² = 64, so √64 = 8
• 9² = 81, so √81 = 9
• 10² = 100, so √100 = 10
• 11² = 121, so √121 = 11
• 12² = 144, so √144 = 12
• 13² = 169, so √169 = 13
• 14² = 196, so √196 = 14
• 15² = 225, so √225 = 15

Once you know these, finding √196 takes half a second. No calculator needed.

2. Long Division Method

This is the old-school way. Slow, but accurate. You won't use it often, but it helps you understand what's actually happening.

Steps:

1. Group digits in pairs from the decimal point moving left and right
2. Find the largest perfect square that fits in your first group
3. Subtract and bring down the next pair
4. Double your current result and figure out what goes next
5. Repeat until you hit your desired precision

It's tedious. That's why nobody does it by hand anymore.

3. Prime Factorization

Break the number into its prime factors. Then pair them up—each pair gives you one factor outside the root.

Example: √72

72 = 2 × 2 × 2 × 3 × 3

Pair the 2s: one pair of 2s gives you a 2 outside

Pair the 3s: one pair of 3s gives you a 3 outside

Lone 2 stays inside

√72 = 2 × 3 × √2 = 6√2

This works great for numbers that are products of perfect squares. For random numbers like 72, you still end up with a radical left over.

4. Calculator

Press the √ button. Done.

This is what 95% of people do. There's no shame in it. Calculators are fast and accurate. If you're doing homework or real work, just use one.

Quick Comparison of Methods

Common Square Root Properties

These rules let you simplify problems without calculating everything out.

Product rule: √(a × b) = √a × √b

So √50 = √(25 × 2) = √25 × √2 = 5√2

Quotient rule: √(a ÷ b) = √a ÷ √b

So √(9/16) = √9 ÷ √16 = 3/4

No addition rule: √(a + b) ≠ √a + √b

This trips people up. √(9 + 16) = √25 = 5. But √9 + √16 = 3 + 4 = 7. Those aren't equal. Don't make this mistake.

How to Get Started: Finding Any Square Root Step by Step

For numbers that aren't perfect squares, here's what to do:

1. Try the nearest perfect square first. For 50, the nearest perfect squares are 49 (7²) and 64 (8²). So √50 is between 7 and 8.
2. Refine with decimals. 7.1² = 50.41. Too high. 7.07² = 49.98. That's close enough for most purposes.
3. Use a calculator for exact values. √50 ≈ 7.0710678...

This estimation method works when you don't have a calculator handy and don't need precision to 10 decimal places.

Square Roots in the Real World

You won't calculate square roots for fun. Here's where they actually show up:

• Construction: Finding diagonal distances. If a room is 12ft by 9ft, the diagonal is √(144 + 81) = √225 = 15ft.
• Statistics: Standard deviation involves square roots.
• Physics: Wave calculations, gravitational formulas, velocity equations.
• Computer graphics: Distance calculations between pixels.

Most people never consciously use square roots after school. But they're baked into the tools and formulas that run everything around you.

Common Mistakes to Avoid

• Forgetting that negative numbers squared become positive—(-5)² = 25, not -25
• Assuming √a + √b = √(a + b)—it doesn't work that way
• Rounding too early in multi-step problems—precision compounds errors
• Confusing √x² with x—it's actually |x| (the absolute value)

Bottom Line

Square roots are simple: find what number, multiplied by itself, gives you the one you started with. Memorize the perfect squares through 144. Know the basic properties. Use a calculator for everything else.

That's it. No need to overthink it.