Simplifying Radicals- Easy Steps and Examples

What Is Simplifying Radicals?

Radicals are just numbers with root symbols. Simplifying them means rewriting them in the simplest form possible—without unnecessary factors trapped under the radical sign.

Here's the bitter truth: unsimplified radicals are harder to work with. They make addition, multiplication, and comparison unnecessarily messy. Once you know the process, you'll spot the pattern in seconds.

Core Vocabulary You Need

• Radical symbol (√) — the square root sign
• Radicand — the number inside the radical
• Index — the small number telling you which root (2 for square root, 3 for cube root, etc.)
• Perfect square — a number whose square root is a whole number (1, 4, 9, 16, 25...)

Most problems you'll see use the square root, which means the index is 2. Teachers often leave that number off because it's implied.

The Golden Rule for Simplifying

This is it—the one rule that handles everything:

√(a × b) = √a × √b

You break the radicand into two factors, pull out whatever is a perfect square, and leave the rest under the radical.

Step-by-Step: How to Simplify Any Radical

Step 1: Find the Prime Factorization

Break the radicand down to its prime factors. Write them out in pairs.

Step 2: Identify the Pairs

For square roots, every pair of identical factors comes out as one single factor outside the radical.

Step 3: Write the Simplified Form

Multiply the outside numbers. Keep remaining unpaired factors under the radical.

Examples That Actually Teach

Example 1: √72

Prime factorization of 72: 2 × 2 × 2 × 3 × 3

Group the pairs: (2 × 2) × 2 × (3 × 3)

Pull out each pair: 2 × 3 × √2

Answer: 6√2

Example 2: √48

Prime factorization of 48: 2 × 2 × 2 × 2 × 3

Group the pairs: (2 × 2) × (2 × 2) × 3

Pull out each pair: 2 × 2 × √3

Answer: 4√3

Example 3: √200

Prime factorization of 200: 2 × 2 × 2 × 5 × 5

Group the pairs: (2 × 2) × 2 × (5 × 5)

Pull out each pair: 2 × 5 × √2

Answer: 10√2

Quick Reference: Common Simplified Forms

Common Mistakes That'll Cost You Points

• Forgetting to factor completely. Stop at prime factors, not just any factors. √50 is not 5√2 because you stopped at 5 × 10. Keep going: 5 × 5 × 2.
• Pulling out factors that aren't perfect squares. √12 is not 2√3. Wait—actually it is. But √18 is not 3√3. See the difference? One factor of 3 stays inside.
• Leaving the radical unsimplified on tests. Teachers notice. If a perfect square factor exists, you're not done.
• Mixing up addition with multiplication. √(4 + 9) does NOT equal √4 + √9. That's not how radicals work. But √(4 × 9) = √4 × √9 works fine.

Getting Started: Your First 5 Practice Problems

Simplify these on your own before checking answers:

1. √36
2. √24
3. √75
4. √128
5. √99

Quick Answers

• √36 = 6 (perfect square)
• √24 = 2√6 (2 × 2 × 2 × 3)
• √75 = 5√3 (3 × 5 × 5)
• √128 = 8√2 (2 × 2 × 2 × 2 × 2 × 2 × 2)
• √99 = 3√11 (3 × 3 × 11)

When You're Ready for More

Once square roots feel automatic, you can move on to:

• Cube roots (index of 3)
• Adding and subtracting radicals
• Rationalizing denominators
• Multiplying radical expressions

Each builds on the same core skill: breaking numbers down and reassembling them cleanly. Master the basics first. Everything else follows.