Simplifying Radicals - Prime Factorization Method Guide

You look at √432 and freeze. The answer isn't obvious like √72 = 6√2. That's where prime factorization saves you — it's the systematic method that always works, no matter how ugly the number.

The Prime Factorization Method — Step by Step

1. Factor the number into primes
2. Pair up identical prime factors
3. Extract one factor from each pair (goes outside the radical)
4. Leave unpaired factors inside the radical

Worked Examples

Example 1: √72

1. 72 = 2 × 2 × 2 × 3 × 3
2. Pairs: (2,2) and (3,3), leftover: 2
3. Extract: 2 × 3 = 6 outside
4. √72 = 6√2

Example 2: √180

1. 180 = 2 × 2 × 3 × 3 × 5
2. Pairs: (2,2) and (3,3), leftover: 5
3. Extract: 2 × 3 = 6 outside
4. √180 = 6√5

Example 3: √432

1. 432 = 2 × 2 × 2 × 2 × 3 × 3 × 3
2. Pairs: (2,2), (2,2), (3,3), leftover: 3
3. Extract: 2 × 2 × 3 = 12 outside
4. √432 = 12√3

Example 4: √2520

1. 2520 = 2³ × 3² × 5 × 7
2. Pairs: (2,2) from 2³, (3,3) from 3², leftover: 2, 5, 7
3. Extract: 2 × 3 = 6 outside
4. Leftover inside: 2 × 5 × 7 = 70
5. √2520 = 6√70

The Shortcut: Largest Perfect Square Method

If you can spot the largest perfect square factor, skip prime factorization:

| Number | Largest Perfect Square Factor | Result | |--------|------------------------------|--------| | √48 | 16 (48 = 16 × 3) | 4√3 | | √75 | 25 (75 = 25 × 3) | 5√3 | | √200 | 100 (200 = 100 × 2) | 10√2 | | √128 | 64 (128 = 64 × 2) | 8√2 | | √245 | 49 (245 = 49 × 5) | 7√5 |

When to use which method:

• Small numbers (< 200): try the shortcut first
• Large numbers or stuck: use prime factorization
• Exam with no calculator: prime factorization is safer

Common Perfect Square Factors Cheat Sheet

| Perfect Square | √ | Perfect Square | √ | |---------------|---|---------------|---| | 4 | 2 | 64 | 8 | | 9 | 3 | 81 | 9 | | 16 | 4 | 100 | 10 | | 25 | 5 | 121 | 11 | | 36 | 6 | 144 | 12 | | 49 | 7 | 169 | 13 |

The Trench Truth: When simplifying radicals on exams, always check your answer by squaring back. If you got √432 = 12√3, verify: (12)² × 3 = 144 × 3 = 432. ✓ If it doesn't match, you made an error in pairing or extraction.

Simplifying Fractions Under Radicals

√(a/b) = √a / √b — split the fraction, simplify each part.

Example: √(18/50)

1. √18 / √50
2. √18 = 3√2, √50 = 5√2
3. 3√2 / 5√2 = 3/5

The radicals cancel! √(18/50) = 3/5. Always simplify before reaching for decimals.

Variables Under Radicals

The same pairing method works with variables:

√(x⁶y³z⁴)

1. x⁶ = (x³)² → extract x³
2. y³ = y² × y → extract y, leave y inside
3. z⁴ = (z²)² → extract z²
4. = x³yz²√y

Rule: even exponents come out as half. Odd exponents come out as (exponent−1)/2 with one left inside.

Use our square root calculator to see the prime factorization and simplification for any number.

Related: Quadratic Formula Calculator · Derivative Calculator · Statistics Calculator