Nth root of a number.
r = x^(1/n)
The nth root of a number x is computed by raising x to the power of 1/n. This works for any positive radicand and any positive integer root index. For even-indexed roots of positive numbers there are two real results (positive and negative), but the principal root is always the positive one. For odd-indexed roots, the real root of a negative number also exists and is negative.
r = 3125^(1/5) = 3125^0.2 = 5
Result: r = 5
The fifth root of 3125 is exactly 5, because 5⁵ = 5 × 5 × 5 × 5 × 5 = 3125. This is a perfect fifth power, so the result is a whole number. For non-perfect powers — for example, the cube root of 10 ≈ 2.1544 — the calculator returns a decimal approximation rounded to four significant figures.
An nth root of a number x is a value r satisfying rⁿ = x. The most familiar case is the square root (n = 2): √9 = 3 because 3² = 9. The cube root (n = 3) and higher roots follow the same logic.
Radical notation writes ⁿ√x with the index n in the notch of the radical symbol. Exponent notation writes x^(1/n). Both are mathematically equivalent and can be used interchangeably in calculators and computer algebra systems.
For even indices and positive radicands, there are always two real roots: one positive and one negative (e.g., √25 = ±5). The principal root is defined as the non-negative root, which is what this calculator returns.
When n is even and x is negative, no real root exists — the result is a complex number (e.g., √(−4) = 2i). This calculator focuses on real-number results.
A number is a perfect nth power when its nth root is a whole integer. Examples: 8 is a perfect cube (∛8 = 2), 81 is a perfect fourth power (⁴√81 = 3).
⁴√625 = 625^(1/4) = 5, because 5⁴ = 625.
Use logarithms: ⁿ√x = e^(ln(x)/n). For simple cases like perfect powers, trial multiplication is faster — e.g., guess and check integers until rⁿ = x.
√2 is irrational. Its decimal expansion is approximately 1.41421356…, continuing forever without repeating. This was one of the first irrational numbers proven by ancient Greek mathematicians.
In compound interest, the formula for per-period rate is r = (FV/PV)^(1/n) − 1, where n is the number of periods. This is a direct application of the nth root.
A radical expression is any algebraic expression that contains a root symbol (√, ∛, etc.) or an exponent of the form 1/n. Simplifying radical expressions is a core skill in algebra.
A square root is simply an nth root where n = 2. The nth root generalises this concept to any positive integer index — cube root (n = 3), fourth root (n = 4), and so on.
Yes. The radicand x can be any non-negative real number, including decimals and fractions. For example, ∛0.125 = 0.5 because 0.5³ = 0.125.
Most nth roots are irrational numbers — they cannot be expressed as a finite decimal or simple fraction. The calculator returns a decimal approximation in those cases, rounded to four significant figures by default.
For odd indices like 3, negative radicands do have a real root. Enter the absolute value, calculate, then apply a negative sign to the result. For example, ∛(−27) = −3 because (−3)³ = −27.
Division by zero occurs in the exponent 1/n, so n = 0 is undefined. The calculator will flag this as an invalid input.
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