How to Do Square Root- Methods and Examples

What Is a Square Root, Anyway?

A square root is the number that, when multiplied by itself, gives you the original number. If √16 = 4, it's because 4 × 4 = 16. That's it. No fancy definitions needed.

You encounter square roots constantly—geometry, physics, finance, coding. Most people panic when they see the √ symbol. They shouldn't. There are four main ways to calculate them, and I'll show you each one.

The Four Methods

1. Prime Factorization

This method works best when you have a perfect square (a number with a whole number square root). You break the number into its prime factors, then pair them up.

Example: √144

Step 1: Factor 144 → 2 × 2 × 2 × 2 × 3 × 3

Step 2: Pair the factors → (2×2) × (2×2) × (3×3)

Step 3: Take one number from each pair → 2 × 2 × 3 = 12

It works. But try this with √50 sometime. You'll get 5√2, which is technically correct but useless if you need a decimal answer.

2. Long Division Method

This is the manual approach that actually works for any number. It's slow and tedious, but accurate.

Example: √121

Step 1: Group digits in pairs from right to left → 1 21

Step 2: Find the largest number whose square fits the first group → 1² = 1. Subtract. Bring down the next pair.

Step 3: Double your current result (1×2=2), then find a digit X where (20+X) × X fits → 21 fits (21×1=21).

Step 4: Subtract → 0

Result: 11

For non-perfect squares like √2, you keep going digit by digit until you've had enough. √2 ≈ 1.41421356... and it never ends.

3. Estimation

Fast. Imperfect. Good enough for real life.

Example: √75

Find the perfect squares around it: 8² = 64, 9² = 81

75 is 11 units above 64 and 6 units below 81.

Divide the gap: 11 ÷ (2 × 8.5) ≈ 0.65

Estimate: 8 + 0.65 = 8.65

Actual answer: 8.660... Close enough for most situations.

4. The Calculator Method

Press √, type the number, hit equals. Done.

This is what 95% of people do. There's no shame in it. Calculators exist for a reason—they're faster and eliminate human error. If you're doing actual math problems, just use the calculator.

Method Comparison

How to Do Square Root: Getting Started

Here's what you actually do in practice:

• Need a quick answer? Use your phone calculator or Google search bar. Type "sqrt(144)" and get 12 instantly.
• Taking a test where calculators aren't allowed? Learn the long division method for common squares (1-20). Memorize 12² = 144, 13² = 169, 15² = 225. You'll spot these fast.
• Estimating in everyday life? Round to the nearest perfect square. √50? That's between 7² (49) and 8² (64), so it's about 7.07.
• Working with algebra? Simplify where possible. √50 becomes 5√2. √72 becomes 6√2. This makes complex equations manageable.

Common Square Root Values to Know

Memorize these. They'll save you time:

• √1 = 1
• √4 = 2
• √9 = 3
• √16 = 4
• √25 = 5
• √36 = 6
• √49 = 7
• √64 = 8
• √81 = 9
• √100 = 10
• √144 = 12
• √169 = 13
• √196 = 14
• √225 = 15

Negative Square Roots

Every positive number has two square roots—one positive, one negative. √16 = 4, but -√16 = -4. Both are correct because 4 × 4 = 16 and (-4) × (-4) = 16.

When you see "±√16" in an equation, it means both 4 and -4 are valid solutions.

Imaginary Numbers

You can't take the square root of a negative number using real numbers. √-1 doesn't exist in the real number system. That's why mathematicians invented i—the imaginary unit where i² = -1.

√-16 = 4i. That's it. Don't panic about this unless you're doing advanced engineering or physics.

Bottom Line

Use a calculator for actual calculations. Learn estimation for quick mental work. Know the prime factorization method for perfect squares. The long division method is mostly academic—you'll never need it in real life unless you're teaching a math class.

Square roots aren't complicated. The symbol is intimidating, but the concept is simple: find what multiplies by itself to get your number.