How-to · Updated October 2, 2026
How do you write a square root in simplest radical form?
Direct answer
Factor the radicand and pull out the largest perfect square. √72 = 6√2 because 72 = 36 × 2 and √36 = 6. The same step gives √24 = 2√6, √12 = 2√3, and √8 = 2√2.
Common square roots
| Radical | Factorization | Simplest form | Decimal |
|---|---|---|---|
| √72 | 2³ × 3² | 6√2 | 8.4853 |
| √24 | 2³ × 3 | 2√6 | 4.899 |
| √12 | 2² × 3 | 2√3 | 3.4641 |
| √8 | 2³ | 2√2 | 2.8284 |
| √45 | 3² × 5 | 3√5 | 6.7082 |
| √50 | 2 × 5² | 5√2 | 7.0711 |
| √180 | 2² × 3² × 5 | 6√5 | 13.4164 |
| √6 | 2 × 3 | √6 | 2.4495 |
| √64 | 2⁶ | 8 | 8 |
6√2 is the exact value. 8.4853 is that number rounded to four decimal places. √64 is a perfect square, so the radical disappears.
The four checks
- No perfect square (or perfect cube, for a cube root) remains under the radical.
- No fraction remains under the radical. √(3/8) = √6/4.
- No radical remains in a denominator. 5/√13 = 5√13/13.
- The index is as small as it can be.
For √180 the factorization is 2² × 3² × 5. The largest perfect square factor is 36, so √180 = 6√5.
Limits
- Square roots of negative numbers use i = √(−1). Odd roots of negatives stay real and keep the minus sign.
- Variables in these simplifications are treated as positive. For an even root, √(x²) is |x|.
- 2√18 is a correct product, and it is not simplest radical form, because 18 still contains a perfect square.
Use the calculator
Combining two of them: how to add and subtract radicals. Enter any radicand in the simplest radical form calculator.