Square roots, cube roots and nth roots, with simplified radicals.
A square root calculator should give more than a long decimal, so this one also gives the simplified radical form: √72 is 8.4853, or 6√2. Switch the root to work out cube roots, 4th, 5th and 6th roots in the same place. For whole numbers it also shows how the radical was simplified, using the number’s prime factors.
How to Use the Square Root Calculator
- Enter a number, such as 72.
- Choose the root: square root (√), cube root (∛), or the 4th, 5th or 6th root.
- Read the decimal result. For whole numbers you also get the simplified radical and the steps.
Even roots of negative numbers are not real numbers, so the calculator shows the result with i, the imaginary unit. Odd roots of negative numbers are real: the cube root of −27 is −3.
What Is a Square Root?
The square root of a number is the value that gives that number when multiplied by itself. √49 = 7 because 7 × 7 = 49. Every positive number has two square roots, one positive and one negative (7 and −7), but the √ sign means the positive one.
A cube root works the same way with three factors: ∛64 = 4 because 4 × 4 × 4 = 64. In general, the nth root of x is the number that gives x when raised to the power n, and it can be written as x^(1/n).
How to Simplify a Square Root
- Break the number into prime factors.
- Group identical factors in pairs (in threes for a cube root).
- Each complete group moves outside the radical as one factor. Anything left over stays inside.
Example: √72 72 = 2 × 2 × 2 × 3 × 3. There is one pair of 2s and one pair of 3s, with a 2 left over. Outside: 2 × 3 = 6. Inside: 2. So √72 = 6√2 ≈ 8.4853.
Worked Examples
- √144 = 12, a perfect square.
- √72 = 6√2 ≈ 8.48528
- ∛54 = 3∛2 ≈ 3.77976, because 54 = 3 × 3 × 3 × 2.
- √50 = 5√2 ≈ 7.07107
- ∛−27 = −3
Perfect Squares to Know
| n | n² | n | n² | |
| 1 | 1 | 9 | 81 | |
| 2 | 4 | 10 | 100 | |
| 3 | 9 | 11 | 121 | |
| 4 | 16 | 12 | 144 | |
| 5 | 25 | 13 | 169 | |
| 6 | 36 | 14 | 196 | |
| 7 | 49 | 15 | 225 | |
| 8 | 64 | 20 | 400 |
Knowing these helps you estimate. √50 must be just over 7, because 7² = 49.
Where Square Roots Show Up
- Room and screen diagonals. By Pythagoras, the diagonal of a 12 ft × 9 ft room is √(12² + 9²) = √225 = 15 ft.
- Squaring a corner. Builders use a 3-4-5 triangle: if two sides are 3 m and 4 m, the diagonal must be √25 = 5 m for a right angle.
- Area to side length. A square plot of 50 m² has sides of √50 ≈ 7.07 m.
Powers, Factors and Next Steps
- To raise a number to a power, such as 5³ or 2^10, use the exponent calculator.
- To see the full list of prime factors the simplification uses, try the prime factorization calculator.
- For fraction answers, the fraction calculator can help with the next step.
Square Root FAQs
What is the square root of 2?
√2 is approximately 1.41421356. It is irrational, so its decimal never ends or repeats, and √2 is already in simplest form.
Why does the calculator show “i” for some answers?
The square root of a negative number is not a real number. It is written using i, where i = √−1. For example, √−16 = 4i.
What does “simplified radical form” mean?
It means no perfect square (or perfect cube, for cube roots) is left inside the radical. 6√2 is simplified, but √72 is not.
How accurate are the decimal answers?
Results are shown to up to 10 significant digits, which is more than enough for schoolwork, engineering estimates and everyday use.
How we checked this page: each root was checked by raising the answer back to the matching power, and the simplified forms were confirmed from their prime factors.