The nth root of a number x is a number y such that y raised to the power of n is equal to x (y^n = x). For example, the square root (2nd root) of 9 is 3, because 3^2 = 9.

Can you take the root of a negative number?

Yes. If the root degree n is odd, the real root of a negative number is negative (e.g. the cube root of -8 is -2). If the degree n is even, the root is an imaginary or complex number (e.g. the square root of -4 is 2i). This calculator supports both real and complex root values.

What is the difference between radical index and degree?

They refer to the same thing: the number n in ⁿ√x. It represents the degree of the root (e.g. 2 for square root, 3 for cube root).

Root Calculator — square root, cube root, nth root

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What is a radical root?

In mathematics, the **nth root** of a number x is a value y that, when multiplied by itself n times, equals x. This is written as: yⁿ = x or y = ⁿ√x.

Square Root (n = 2): Finding a number that multiplied by itself once equals the radicand. E.g. √49 = 7 since 7 × 7 = 49.
Cube Root (n = 3): Finding a number that multiplied by itself twice equals the radicand. E.g. ³√125 = 5 since 5 × 5 × 5 = 125.
Higher and Fractional Roots: Any positive number can serve as the root degree. For fractional degrees, the calculation is defined as x^(1/n).

Negative numbers and imaginary roots

The behavior of roots when the radicand is negative depends entirely on whether the root degree is odd or even:

Odd Degrees: Negative numbers have a real, negative root. For instance, the cube root of -27 is -3, because (-3) × (-3) × (-3) = -27.
Even Degrees: Even roots of negative numbers cannot be represented by real numbers (since any real number squared or raised to an even power is non-negative). Instead, they are represented as **complex numbers** containing the imaginary unit i (where i² = -1). For example, the square root of -16 is 4i.

Units, rounding, and limits

This calculator operates on high-precision BigNumber arithmetic with up to 14 decimal digits. When taking roots that result in infinite repeating or non-terminating decimals (like √2), the result is rounded to 14 decimal places. Complex numbers are formatted as a + bi.

FAQ

What does radical index mean?

The index of a radical is the number of times the root must be multiplied by itself to equal the radicand. It is the degree of the root (the n in ⁿ√x).

Can the root degree be a decimal?

Yes, the root degree can be any positive real number. For example, the 2.5th root of 32 is 4, because 4^2.5 = 32.

Is the square root of a number always positive?

In arithmetic, the symbol √ refers specifically to the **principal square root**, which is the non-negative root of a non-negative number. While (-3)² = 9 as well, the principal root is √9 = 3.

Disclaimer. RapidRatio is informational only. It is not financial, tax, business, or professional advice. Verify results and assumptions with qualified professionals before making decisions.