Square & Cube Root Calculator
The square root of a number x is a value r such that r² = x. More generally, the nth root of x is the value r satisfying rⁿ = x, computed as x^(1/n). This calculator instantly finds the square root, cube root, and any custom nth root of any real number, and automatically detects whether the result is a perfect square, perfect cube, or perfect power.
Negative numbers have no real square root (or any even-degree root), but do have real cube roots and other odd-degree roots. Enter your number and optionally adjust the root degree to explore all three results side by side. Perfect-square and perfect-cube badges highlight exact integer results.
How it works
ⁿ√x = x^(1/n). Square root: √x = x^(1/2). Cube root: ∛x = x^(1/3). For negative x and even n, the result is not real. A number x is a perfect nth power when x^(1/n) is an integer.
Use cases
- Finding the side length of a square given its area
- Solving quadratic and higher-degree polynomial equations
- Computing standard deviation in statistics (involves square roots)
- Determining the edge length of a cube given its volume
- Simplifying radical expressions in algebra homework
Frequently asked questions
What is the square root of a number?
The square root of a number x is a value r such that r² = x. For example, the square root of 25 is 5, because 5 × 5 = 25. Every positive number has two square roots, one positive and one negative, but the calculator reports the principal (positive) root.
How do I find the cube root of a number?
The cube root is the value r that satisfies r³ = x, computed as x^(1/3). For example, the cube root of 27 is 3, because 3 × 3 × 3 = 27. Unlike square roots, cube roots are defined for negative numbers too: the cube root of −8 is −2.
Can you take the square root of a negative number?
Not within the real numbers. A negative number has no real square root, or any even-degree root, because no real value squared gives a negative result. Odd-degree roots such as the cube root do exist for negatives. The calculator flags even roots of negative inputs as not real.
What is a perfect square?
A perfect square is a number whose square root is a whole integer, such as 1, 4, 9, 16, 25, and 36. Likewise a perfect cube has an integer cube root, like 8, 27, and 64. The calculator displays a badge whenever your result is an exact perfect square, cube, or nth power.
How do I calculate the nth root of a number?
The nth root of x equals x^(1/n), the value r that satisfies rⁿ = x. For example, the 4th root of 81 is 3, because 3⁴ = 81. Enter your number and set the root degree n to compute any root beyond square and cube.