Root Calculator

Roots are exponents in disguise—√x is x^(1/2)—but negative radicands and even degrees pull you into complex numbers fast. I still sketch a tiny number line when students ask why √−4 is not −2. This calculator defaults to square root when you leave the degree blank and flags complex results in plain a ± bi language when reals will not do.

How to Use This Root Calculator

• Enter the radicand (x). The value under the radical—64, −8, 2.25, whatever you need.
• Enter the root degree (n). 2 for square root, 3 for cube root, or any positive real. Leave blank to assume 2.
• Press Calculate. Real results print as decimals; complex results show a + bi style text when the even root of a negative requires it.
• Read the subtitle. It names the operation (√, cube root, or nth root) and notes when the answer is complex.

Fractional exponents overlap this math—see the Exponent Calculator for x^(1/n) written as powers. For logarithmic inverses, try the Log Calculator.

Root Formulas and Practical Applications

The nth root undoes raising to the nth power—like asking which edge length cubes to fill a box of known volume.

y = ⁿ√x ⟺ yⁿ = x

Computationally, ⁿ√x = x^(1/n). Square roots appear in Pythagorean geometry; cube roots in volume scaling; higher roots in engineering tolerance specs.

Square and cube roots

√49 = 7 because 7² = 49. ³√125 = 5 because 5³ = 125. Leave n empty for squares; type 3 for cubes.

Negative radicands

Odd integer n keeps answers real: ³√(−27) = −3. Even integer n with negative x yields complex values—√(−16) = 4i in principal form. The tool chooses the real root for odd integers and complex polar construction for even roots of negatives.

Non-integer degrees

The 2.5th root of 32 is 4 because 4^2.5 = 32. Decimal degrees model fractional-power scaling in finance and physics—handled via x^(1/n) with high-precision arithmetic.

Leave the degree field empty and the calculator assumes n = 2—square root. That default matches how most people reach for the √ key first. Results format with up to 14 decimal places; non-terminating roots like √2 show a trimmed decimal rather than an endless tail.

Complex outputs use a + bi style text when even roots of negative radicands require imaginary units. Odd integer roots of negatives stay on the real line—cube root of −8 is −2, not a complex value—matching what most algebra courses emphasize before polar form.

Roots and exponents are two names for the same operation: ⁿ√x equals x^(1/n). If you already rewrote a problem with a caret, the Exponent Calculator accepts the same numbers. Square roots appear in the Pythagorean Theorem Calculator when you solve for a missing leg or hypotenuse length.

Validation rejects zero or negative degrees because dividing 1 by zero breaks the exponent form, and non-positive indices are outside this tool’s scope. When in doubt, leave the degree blank for a square root and add n = 3 explicitly for cube roots—most inventory and geometry quick checks stop there.

Try √2 and √3 when calibrating expectations—non-terminating decimals should end at a clean trimmed tail, not an endless random-looking string from binary float noise. Complex results like √(−4) → 2i follow the principal branch convention used in most algebra courses.

Radicand zero with positive degree n returns zero—the nth root of 0 is 0. Positive radicands always work for positive degrees; the tricky cases are negative radicands with even versus odd index, where real versus complex paths diverge.

Frequently Asked Questions

Read the subtitle under the primary result—it names square, cube, or nth root and flags complex output when imaginary units appear. For geometry checks, compare √area when you know a square’s area, or ∛volume when a cube’s volume is given. Principal roots are non-negative for non-negative radicands even though equations like x² = 9 have two solutions.

What is an nth root?

A value that reproduces x when raised to the nth power. Index, degree, and radical order all name the same n in ⁿ√x.

Can you take the root of a negative number?

Sometimes. Odd integer roots stay real and negative; even integer roots of negatives become complex. Positive radicands always work for positive degrees.

What is the default root degree?

Blank degree field → n = 2. Matches the everyday √ symbol most people reach for first.

Can the root degree be a decimal?

Yes—any positive real n. Validation rejects zero or negative degrees because 1/0 and even-root ambiguities break down.

Is the square root always the positive root?

For real x ≥ 0, the principal √ is non-negative. (−3)² also equals 9, but √9 = 3 in the convention this tool uses for real outputs.

Disclaimer. RapidRatio is informational only—not engineering sign-off. Verify critical root calculations with appropriate domain software.