Square Root Calculator - Complete Guide to Radicals

You can punch √72 into a calculator and get 8.485. But on exams, in proofs, and in higher math, you need the exact form: 6√2. This guide covers everything about radicals — from simplification to operations.

Simplifying Radicals — The Core Skill

Rule: √(a·b) = √a · √b

Extract the largest perfect square factor from under the radical.

Step-by-Step Method

1. Factor the number under the radical
2. Identify perfect square factors
3. Pull perfect squares out of the radical
4. Multiply coefficients outside

Examples

| Input | Factor | Pull Out | Result | |-------|--------|----------|--------| | √12 | 4 × 3 | √4 = 2 | 2√3 | | √27 | 9 × 3 | √9 = 3 | 3√3 | | √45 | 9 × 5 | √9 = 3 | 3√5 | | √75 | 25 × 3 | √25 = 5 | 5√3 | | √98 | 49 × 2 | √49 = 7 | 7√2 | | √200 | 100 × 2 | √100 = 10 | 10√2 |

Prime Factorization for Hard Cases

When the perfect square factor isn't obvious:

√180:

1. 180 = 2² × 3² × 5
2. Pair up: (2²) and (3²) come out as 2 and 3
3. Leftover: 5
4. √180 = 2 × 3 × √5 = 6√5

√252:

1. 252 = 2² × 3² × 7
2. Pairs: (2²) and (3²) → 2 × 3 = 6
3. Leftover: 7
4. √252 = 6√7

Adding and Subtracting Radicals

Rule: Only like radicals (same number under √) can be combined.

| Expression | Like Radicals? | Result | |-----------|---------------|--------| | 3√2 + 5√2 | Yes (both √2) | 8√2 | | 2√3 − 7√3 | Yes (both √3) | −5√3 | | 4√2 + 3√5 | No | 4√2 + 3√5 (can't combine) | | √8 + √18 | Simplify first! | 2√2 + 3√2 = 5√2 |

The Trench Truth: The #1 mistake with radical addition: trying to combine unlike radicals. √2 + √3 ≠ √5. Always simplify first — what looks unlike (√8 + √18) might become like (2√2 + 3√2).

Multiplying Radicals

√a · √b = √(ab)

| Expression | Calculation | Result | |-----------|------------|--------| | √3 · √5 | √15 | √15 | | √2 · √8 | √16 | 4 | | 2√3 · 4√3 | 8 · 3 | 24 | | √6 · √6 | √36 | 6 |

Rationalizing the Denominator

Fractions shouldn't have radicals in the denominator. Multiply top and bottom by the radical:

Single Radical

3/√5 → 3√5/5

Multiply by √5/√5: (3·√5)/(√5·√5) = 3√5/5

Binomial Radical (Conjugate Method)

1/(√3 + √2) → (√3 − √2)/1 = √3 − √2

Multiply by conjugate (√3 − √2)/(√3 − √2):

Numerator: 1·(√3 − √2) = √3 − √2

Denominator: (√3)² − (√2)² = 3 − 2 = 1

Square Root Operations Summary

| Operation | Rule | Example | |-----------|------|---------| | Simplify | Extract perfect squares | √72 = 6√2 | | Add/Subtract | Combine like radicals only | 3√2 + √2 = 4√2 | | Multiply | √a · √b = √(ab) | √3 · √7 = √21 | | Divide | √a / √b = √(a/b) | √20/√5 = √4 = 2 | | Rationalize | Multiply by conjugate | 1/(√5+1) = (√5−1)/4 |

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Related: Quadratic Formula Calculator · Derivative Calculator · Integral Calculator