Square root calculator
Square roots, cube roots and nth roots to 12 significant digits, with the simplified radical form.
Perfect squares and cubes
| n | n² | n³ | √n |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 1.41421 |
| 3 | 9 | 27 | 1.73205 |
| 4 | 16 | 64 | 2 |
| 5 | 25 | 125 | 2.23607 |
| 6 | 36 | 216 | 2.44949 |
| 7 | 49 | 343 | 2.64575 |
| 8 | 64 | 512 | 2.82843 |
| 9 | 81 | 729 | 3 |
| 10 | 100 | 1,000 | 3.16228 |
| 11 | 121 | 1,331 | 3.31662 |
| 12 | 144 | 1,728 | 3.4641 |
| 13 | 169 | 2,197 | 3.60555 |
| 14 | 196 | 2,744 | 3.74166 |
| 15 | 225 | 3,375 | 3.87298 |
| 16 | 256 | 4,096 | 4 |
| 17 | 289 | 4,913 | 4.12311 |
| 18 | 324 | 5,832 | 4.24264 |
| 19 | 361 | 6,859 | 4.3589 |
| 20 | 400 | 8,000 | 4.47214 |
Roots are fractional powers
ⁿ√x = x^(1/n). So √x = x^0.5 and ∛x = x^(1/3). This is why the exponent rules apply to roots: √(ab) = √a·√b, and √(x²) = |x|.
More on powers in the exponent calculator.
Show me the steps
An AI tutor walks through the working for this exact result. The answer above is computed in your browser; the explanation is optional and is the only thing that leaves your device.
Find the square root of 72 and simplify the radical = 8.48528137424; 6√2
Questions people ask
How do I simplify a square root?
Find the largest perfect square that divides the number and take its root outside. √72 = √(36 × 2) = 6√2. The calculator does this for any whole number and any root index.
Can you take the square root of a negative number?
Not in the real numbers, because any real number squared is zero or positive. In complex numbers √(−9) = 3i. Odd roots of negatives are fine: ∛(−8) = −2.
Why does √9 give 3 and not ±3?
The √ symbol means the principal (non-negative) root, so √9 = 3. The equation x² = 9 has two solutions, x = 3 and x = −3, which is why you write x = ±√9 when solving.
How do I estimate a square root by hand?
Find the perfect squares either side: √50 lies between √49 = 7 and √64 = 8, closer to 7. Then refine with Newton’s method: guess g, replace it with (g + 50/g)/2. From 7: (7 + 50/7)/2 ≈ 7.0714, already correct to 4 digits (√50 ≈ 7.07107).