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Calculate nth roots (ⁿ√x) for any degree. Find cube roots, fourth roots, fifth roots, and more. Handles negative radicands for odd roots with complete step-by-step solutions.
Enter values to calculate the nth root
Enter the number you want to find the root of (the radicand). This can be positive or negative (negative numbers work with odd roots only).
Specify which root you want to calculate. Enter 2 for square root, 3 for cube root, 4 for fourth root, and so on. Must be an integer ≥ 2.
The calculator shows the simplified form, decimal approximation, and step-by-step solution including prime factorization and simplification process.
ⁿ√x = x^(1/n)
The nth root of x is the number that, when raised to the power n, gives x.
ⁿ√(ab) = ⁿ√a × ⁿ√b
Example: ³√(8×27) = ³√8 × ³√27 = 2 × 3 = 6
ⁿ√(a/b) = ⁿ√a / ⁿ√b
Example: ³√(64/8) = ³√64 / ³√8 = 4 / 2 = 2
ⁿ√(xᵐ) = (ⁿ√x)ᵐ
Example: ³√(2³) = (³√2)³ = 2
ⁿ√(-x) = -ⁿ√x (odd n only)
Example: ³√(-8) = -2 (valid for odd roots)
ⁿ√(xᵐ) = x^(m/n)
Example: ³√(x²) = x^(2/3)
| Expression | Result | Type |
|---|---|---|
| ³√8 | 2 | Cube root |
| ³√27 | 3 | Cube root |
| ³√64 | 4 | Cube root |
| ³√125 | 5 | Cube root |
| ³√-8 | -2 | Cube root (negative) |
| ⁴√16 | 2 | Fourth root |
| ⁴√81 | 3 | Fourth root |
| ⁴√256 | 4 | Fourth root |
| ⁵√32 | 2 | Fifth root |
| ⁵√243 | 3 | Fifth root |
| ⁶√64 | 2 | Sixth root |
| ⁶√729 | 3 | Sixth root |
An nth root of a number x is a value that, when raised to the power n, equals x. For example, the 3rd root (cube root) of 27 is 3 because 3³ = 27. The notation is ⁿ√x = y where yⁿ = x.
To calculate ⁿ√x, use the formula x^(1/n). For example, ³√8 = 8^(1/3) = 2. You can also use prime factorization: break x into prime factors, then for every n identical factors, take one outside the radical.
You can only take odd roots (3rd, 5th, 7th, etc.) of negative numbers. For example, ³√(-8) = -2 is valid. Even roots (2nd, 4th, 6th, etc.) of negative numbers produce complex numbers, not real numbers.
A square root (²√) asks 'what number squared equals x?', while a cube root (³√) asks 'what number cubed equals x?'. Square root is n=2, cube root is n=3. For example, √16 = 4 (because 4² = 16) and ³√27 = 3 (because 3³ = 27).
Use prime factorization. For ⁿ√x, find all prime factors of x. For every n identical factors, move one outside the radical. For example, to simplify ³√54: 54 = 2×3³, so ³√54 = ³√(2×27) = 3³√2.
Perfect nth powers are numbers that have exact integer nth roots. Perfect cubes include 1, 8, 27, 64, 125 (1³, 2³, 3³, 4³, 5³). Perfect fourth powers include 1, 16, 81, 256 (1⁴, 2⁴, 3⁴, 4⁴). These roots simplify to whole numbers.
Nth roots generalize the concept of square roots and cube roots to any positive integer degree. While square roots ask "what number times itself gives x?", nth roots ask "what number raised to the nth power gives x?" This makes nth roots the inverse operation of exponentiation.
The notation ⁿ√x is equivalent to x^(1/n), connecting radicals to fractional exponents. This connection allows us to use exponential rules to manipulate and simplify radical expressions. Understanding nth roots is essential for algebra, calculus, and advanced mathematics.