Radical Rules- Simplifying Square Roots

What Is a Square Root, Really?

A square root is just the opposite of squaring a number. When you square 5, you get 25. When you take the square root of 25, you get 5. That's it. No magic, no mystery.

The radical symbol (√) tells you to find the number that, when multiplied by itself, gives you the number inside. √36 = 6 because 6 × 6 = 36.

Most numbers don't have nice clean square roots. √20 isn't a whole number. That's where simplifying comes in—you break down the mess into cleaner parts.

The Core Rule You Need to Know

Product Rule for Radicals:

√(a × b) = √a × √b

This single rule solves almost every simplification problem. You split the number under the radical into factors, pull out the perfect squares, and leave the rest inside.

Quotient Rule:

√(a ÷ b) = √a ÷ √b

Same idea. Divide first, then simplify—or simplify first, then divide. Either way works.

How to Simplify Square Roots: Step by Step

Here's the process that actually works:

1. Factor the number under the radical into prime factors
2. Find all the perfect square factors (1, 4, 9, 16, 25, 36, 49, 64, 81, 100...)
3. Pull those perfect squares outside the radical
4. Multiply what you've pulled out
5. Leave the non-perfect-square factors inside

Example: Simplify √72

Step 1: Factor 72

72 = 8 × 9 = 4 × 2 × 9 = 36 × 2

Step 2: Pull out the perfect square

√72 = √(36 × 2) = √36 × √2 = 6√2

Done. 6√2 is simpler than √72. Both equal the same thing, but 6√2 is cleaner.

Example: Simplify √48

48 = 16 × 3

√48 = √16 × √3 = 4√3

That's your answer.

Example: Simplify √200

200 = 100 × 2

√200 = √100 × √2 = 10√2

See the pattern? Find the biggest perfect square that divides evenly, break it out, and you're done.

Perfect Squares to Memorize

You need these memorized or instantly recognizable. They come up constantly.

Memorize up to 15² at minimum. If you're comfortable with 20² (400), even better.

Simplifying Radicals with Variables

Variables work the same way. √(x²) = x. √(x³) = x√x. √(x⁴) = x².

Take √(18x³):

18 = 9 × 2, and x³ = x² × x

√(18x³) = √(9 × 2 × x² × x) = √9 × √x² × √(2x) = 3x√(2x)

The rule doesn't change. Find the perfect squares, pull them out.

Adding and Subtracting Simplified Radicals

This trips people up. You can only add or subtract radicals when the simplified form is identical.

3√2 + 5√2 = 8√2 ✅ Same radical, combine them

3√2 + 5√3 = 3√2 + 5√3 ❌ Different radicals, can't combine

3√2 + 5√2 + 2√3 = 8√2 + 2√3 ✅ Combine like terms only

Think of it like algebra with variables. You can combine 3x + 5x, but not 3x + 5y.

Multiplying Radicals

Multiply the parts outside together, then multiply the parts inside together.

(3√2)(4√3) = 12√6

(2√5)(6√5) = 12 × 5 = 60 ✅ Because √5 × √5 = 5

When you multiply a radical by itself, you get the number underneath. That's squaring in reverse.

Rationalizing the Denominator

Sometimes you'll get √2/2 and need to clean up the bottom. Multiply top and bottom by the radical in the denominator.

1/√2 × √2/√2 = √2/2

The value doesn't change—you just multiplied by 1—but now there's no radical in the denominator. Some teachers require this. Check what your problem set expects.

Getting Started: Your Quick Practice Routine

Pick 10 numbers. Simplify them. Check your answers with a calculator (√n should equal your simplified form squared). Do this daily for a week and it becomes automatic.

Start with these:

• √18
• √32
• √45
• √50
• √75
• √98
• √108
• √128
• √162
• √200

Answers: 3√2, 4√2, 3√5, 5√2, 5√3, 7√2, 6√3, 8√2, 9√2, 10√2

If you got those right without peeking, you understand the process.

Common Mistakes to Avoid

√(a + b) ≠ √a + √b. This is wrong. The radical distributes over multiplication only, not addition.

√(9 + 16) = √25 = 5. But √9 + √16 = 3 + 4 = 7. Those aren't the same. Don't make this mistake.

Another one: forgetting to factor completely. √50 isn't 5√2 is wrong. √50 = √(25 × 2) = 5√2. That's right. The 5 comes from √25, not from guessing.

One more: leaving perfect squares inside the radical. √72 = 6√2, not 6√4. √4 = 2, so 6√4 = 12, which is wrong. Pull everything out that you can.

When You're Stuck

Factor the number completely. Write out all the prime factors. Circle the pairs. Whatever you can pair up comes outside as a single number. Whatever's left unpaired stays inside.

72 = 2 × 2 × 2 × 3 × 3

Pairs: 2×2 and 3×3. That's 2 and 3 outside. Leftover: one 2 inside.

Result: 2 × 3 × √2 = 6√2

This method never fails. Use it when you're unsure.