Root Calculator: Square Root, Cube Root, and Nth Root
Enter any number and choose the root degree to find its square root, cube root, or any nth root. Results update instantly and include a step-by-step estimation walkthrough, a check for whether the result is a perfect root, the equivalent exponent form, and the inverse (the radicand raised to the root degree). Negative radicands are supported when the root degree is odd.
What is a root?
The nth root of a number a is a value b such that b raised to the nth power equals a. Written as the radical symbol with a small n in the index position, or equivalently as a^(1/n) in exponent form. The most familiar roots are the square root (n = 2) and the cube root (n = 3), but any positive whole number n is valid. For example, the cube root of 64 is 4 because 4 x 4 x 4 = 64.
How to calculate any nth root
Every root can be expressed as a fractional exponent: the nth root of a equals a^(1/n). You can compute this on any scientific calculator using the exponent key. For manual estimation, start with any reasonable guess b, then refine it by computing [b x (n-1) + a/b^(n-1)] / n and repeating until the result stabilises to the desired precision. This is the generalised Newton method and it converges quickly even for large root degrees.
Square roots and cube roots
Square roots (n = 2) are the most common. The square root of a perfect square such as 4, 9, 16, 25, 36, 49, 64, 81, or 100 is an exact integer. For non-perfect squares the result is irrational, meaning it goes on forever without repeating. Cube roots (n = 3) work similarly: the cube root of 8 is 2, of 27 is 3, of 64 is 4. Unlike square roots, cube roots of negative numbers are real: the cube root of -8 is -2 because (-2)^3 = -8.
Negative numbers and even vs odd roots
The sign of the radicand matters for even-degree roots. A square root (n = 2) of a negative number has no real solution because no real number squared gives a negative result. The same applies to any even n. For odd root degrees (3, 5, 7...) the real nth root of a negative number exists and is negative: for instance, the 5th root of -32 is -2 because (-2)^5 = -32. This calculator handles both cases and flags even-root negatives clearly.
Perfect roots
A number is a perfect nth power if its nth root is an exact integer. Perfect squares include 1, 4, 9, 16, 25, ... Perfect cubes include 1, 8, 27, 64, 125, 216, ... When you calculate the root of a perfect power, the result carries no rounding error, which makes it useful in algebra, number theory, and computing. The reference table below lists common perfect squares, cubes, and fourth powers up to base 25.
Common perfect roots
| Base (b) | Square (b²) | Cube (b³) | Fourth power (b⁴) |
|---|---|---|---|
| 1 | 1 | 1 | 1 |
| 2 | 4 | 8 | 16 |
| 3 | 9 | 27 | 81 |
| 4 | 16 | 64 | 256 |
| 5 | 25 | 125 | 625 |
| 6 | 36 | 216 | 1296 |
| 7 | 49 | 343 | 2401 |
| 8 | 64 | 512 | 4096 |
| 9 | 81 | 729 | 6561 |
| 10 | 100 | 1000 | 10000 |
| 12 | 144 | 1728 | 20736 |
| 15 | 225 | 3375 | 50625 |
| 20 | 400 | 8000 | 160000 |
| 25 | 625 | 15625 | 390625 |
Integer bases and their squares, cubes, and fourth powers, which produce perfect roots.
Frequently asked questions
How do I find the square root of a number?
Set the root degree to 2 and enter your number. The calculator gives the square root instantly. For a perfect square such as 144 the result is a whole number (12); for other numbers such as 2 the result is an irrational decimal (1.4142...). You can also use the exponent form: the square root of a equals a^0.5.
Can I take the square root of a negative number?
Not as a real number. No real number multiplied by itself gives a negative result, so the square root of a negative number does not exist on the real number line. In complex mathematics it is expressed using the imaginary unit i, so the square root of -9 is 3i. For odd root degrees such as cube root, negative numbers do have real results: the cube root of -27 is -3.
What is the cube root of a number?
The cube root of a number a is a value b such that b x b x b = a. Set the root degree to 3. Common exact cube roots: cube root of 8 = 2, cube root of 27 = 3, cube root of 125 = 5, cube root of 1000 = 10. Cube roots of negative numbers are also real: cube root of -64 = -4.
What does nth root mean?
The nth root of a number a is the value b where b^n = a. The word "degree" or "index" refers to n. Square roots have degree 2, cube roots degree 3, fourth roots degree 4, and so on. Higher-degree roots shrink rapidly: the 10th root of 1024 is 2 because 2^10 = 1024.
How do I write a root as an exponent?
The nth root of a equals a^(1/n). For example, the square root is a^(1/2) or a^0.5, the cube root is a^(1/3), the fourth root is a^(1/4). In a spreadsheet you can write =A1^(1/3) to compute the cube root of the value in cell A1.
What is a perfect square or perfect cube?
A perfect square is an integer that is the square of another integer: 1, 4, 9, 16, 25, 36... A perfect cube is an integer that is the cube of another integer: 1, 8, 27, 64, 125... Their roots are exact whole numbers with no rounding. This calculator shows whether your input is a perfect nth power.
Why does this calculator show decimal places options?
Most roots are irrational numbers that continue forever. The decimal places option lets you choose how much precision you need: 0 places for a rough integer estimate, 4 for everyday work, 8 or 10 for scientific and engineering use. The result is rounded only for display; the underlying computation uses full floating-point precision.