Quick Answer
A square root calculator with steps finds the number that, multiplied by itself, equals your input — written √x, or the nth root ⁿ√x for higher degrees. This tool returns the decimal value, the simplified radical form (e.g. √72 = 6√2), and the full prime-factorization working.
Square Root Formula & Simplified Radical Form
The square root of x is the value y where y² = x. The general nth root extends this: ⁿ√x = x1/n. To simplify, factor the radicand and pull out any factor that repeats n times.
√x = y such that y² = x · ⁿ√x = x1/n. Example: √72 = √(36 × 2) = 6√2.
| Number (n) | √n (decimal) | Simplified radical |
|---|---|---|
| 8 | 2.828 | 2√2 |
| 12 | 3.464 | 2√3 |
| 18 | 4.243 | 3√2 |
| 50 | 7.071 | 5√2 |
| 72 | 8.485 | 6√2 |
How to Use the Square Root Calculator
- Type your number
Enter the value you want the root of (the radicand), for example 72. Decimals and negative numbers work too.
- Pick the root degree
Tap √ for square root, ∛ for cube root, ∜ for fourth root, or type a custom index n (up to 20).
- Read the answer
The decimal value, the simplified radical form, and the step-by-step prime factorization update instantly as you type.
Worked Examples: Square, Cube & Nth Roots
| Input | Root | Result | Working |
|---|---|---|---|
| 72 | Square (√) | 6√2 ≈ 8.485 | 72 = 2³ × 3² → extract 3² and one 2: √(36 × 2) = 6√2 |
| 54 | Cube (∛) | 3∛2 ≈ 3.780 | 54 = 2 × 3³ → extract 3³: ∛(27 × 2) = 3∛2 |
| −9 | Square (√) | 3i | Negative under an even root → imaginary: √−9 = √9 · i = 3i |
A perfect square like 49 returns a whole number (7) with no radical; a number with no repeated factors (like 7) stays under the root as √7.
