What a “root” means (and why it’s everywhere)

A root answers a simple question: “What number do I multiply by itself (a certain number of times) to get this value?” The most common root is the square root, written as √x. It’s the number which, when squared, gives x. But roots come in all “degrees”:

• Square root: √x is the same as x^1/2.
• Cube root: ∛x is the same as x^1/3.
• n-th root: ⁿ√x is the same as x^1/n.

In general, the n-th root of a number a is a value r such that: rⁿ = a. If you think of exponents as “repeat multiplication,” roots are the reverse operation: they “undo” an exponent.

The most important relationship is: ⁿ√a = a^1/n. That means you can compute roots using exponent rules, and you can simplify many expressions by converting roots to fractional exponents. For example:

• √16 = 16^1/2 = 4
• ∛27 = 27^1/3 = 3
• ⁴√81 = 81^1/4 = 3 (because 3⁴ = 81)

Roots behave differently for negative numbers:

• Odd roots of negatives are real. Example: ∛(-8) = -2 because (-2)³ = -8.
• Even roots of negatives are not real. Example: √(-9) has no real solution. In complex numbers, √(-9) = 3i.

This calculator supports both cases. If you enter a negative value with an even degree, you can still get a correct complex root (and optionally all n complex roots).

How this Roots Calculator computes your answer

This tool focuses on accuracy and clarity. Here’s the exact logic:

• It reads your input number a and root degree n.
• If a ≥ 0, it computes the principal real root: a^1/n.
• If a < 0 and n is an odd integer, it returns a real negative root: -|a|^1/n.
• If a < 0 and n is even, the real root doesn’t exist, so it returns the principal complex root using Euler’s form.

Any nonzero number has exactly n complex n-th roots. In polar form, write the number as: a = r·(cos θ + i sin θ). Then the roots are: r^1/n · (cos((θ + 2πk)/n) + i sin((θ + 2πk)/n)) for k = 0, 1, …, n−1. If you turn on “Show all complex roots,” the calculator will list them.

Results are rounded to a reasonable number of decimals by default. If you need more precision, use the “Decimals” option. For exact radicals (like √50 = 5√2), you’d need symbolic simplification. This page focuses on fast numeric answers that work for homework checks, engineering estimates, finance formulas, and everyday math.

Roots examples you can copy-paste

Try these to see how roots behave across different cases:

• Input: a = 144, n = 2
• Result: √144 = 12
• Check: 12² = 144

• Input: a = -125, n = 3
• Result: ∛(-125) = -5
• Check: (-5)³ = -125

• Input: a = 625, n = 4
• Result: ⁴√625 = 5
• Check: 5⁴ = 625

• Input: a = -16, n = 2
• Result: √(-16) = 4i (principal root)
• All square roots: 4i and -4i

If you know the area of a square and want the side length, you take a square root. If a square has area 81, the side length is √81 = 9. Similarly, if you know the volume of a cube and want the side length, you take a cube root. A cube with volume 216 has side length ∛216 = 6.

Frequently Asked Questions

• What is the difference between √x and ⁿ√x?

  √x is the square root (n = 2). ⁿ√x is the n-th root for any degree n, like cube roots (n = 3) or 4th roots (n = 4). They’re all special cases of a^1/n.

• Why does √(-1) equal i?

  Because no real number squared equals -1. Complex numbers extend the number system with i defined as the solution to i² = -1. That’s why √(-9) = 3i and √(-16) = 4i (principal roots).

• Does every number have multiple roots?

  In complex numbers, yes. Any nonzero number has exactly n complex n-th roots. For real numbers, you often talk about the principal root (the standard one shown by calculators), but there are multiple solutions in the complex plane.

• What does “principal root” mean?

  The principal root is the single root chosen by convention so calculators always give one consistent answer. For positive real numbers, it’s the positive real root. For complex cases, it’s the root with angle in the principal range.

• Can you simplify radicals like √50?

  This calculator is optimized for numeric roots. For √50, the decimal result is about 7.0711. Symbolically, √50 = √(25·2) = 5√2. If you need symbolic simplification often, use the “Square Root Calculator” and “Prime Factorization” tools in the links below.

• How do roots relate to exponents and logarithms?

  Roots are fractional exponents: ⁿ√a = a^1/n. Logarithms are another inverse of exponents: log_b(a) answers “what exponent on b gives a?” These three topics—exponents, roots, logs—are a connected triangle.

• Is this Roots Calculator accurate for homework and engineering?

  Yes for numeric answers. It uses the same math most scientific calculators use for principal roots, and it can show complex roots. For formal proofs or exact radical simplification, you may want symbolic algebra software, but for checking and estimating, this is fast and reliable.

• What’s a quick way to sanity-check a root?

  Raise the answer back to the power n. If you computed r = ⁿ√a, then rⁿ should equal a (or be extremely close due to rounding). This page shows a “Check” value so you can verify instantly.

Root properties (the cheat-sheet you actually use)

Once you understand ⁿ√a as a^1/n, a lot of “root tricks” become simple exponent rules. These identities are the ones you’ll see in algebra, geometry, physics, coding, and finance:

ⁿ√(ab) = ⁿ√a · ⁿ√b (works cleanly when a and b are nonnegative reals; in complex numbers there are extra caveats). Example: √(36·4) = √36 · √4 = 6·2 = 12.

ⁿ√(a/b) = ⁿ√a / ⁿ√b (again, typically used for nonnegative real values). Example: √(49/4) = √49 / √4 = 7/2 = 3.5.

(a^m)^1/n = a^m/n. Example: ³√(2⁶) = 2^6/3 = 2² = 4.

If a number has a perfect n-th power factor, you can “pull it out.” Example: √72 = √(36·2) = 6√2. Another: ³√(54) = ³√(27·2) = 3³√2.

The “odd = real, even = complex” rule is the big one: ⁿ√(-a) is real only when n is odd. When n is even, you need complex numbers to represent the solution.

Pro tip: If you’re simplifying by hand, prime factorization is the fastest way to spot perfect squares/cubes. That’s why the “Prime Factorization” tool in the Related section pairs well with roots.

Quick fixes for the most common root errors

• Mistake: √(a + b) = √a + √b.
  Fix: This is not true in general. Example: √(9+16)=√25=5, but √9+√16=3+4=7.
• Mistake: Forgetting the “±” for square roots when solving equations.
  Fix: If x² = 25, then x = ±5. (The symbol √25 usually means the principal root 5.)
• Mistake: Treating even roots of negatives as real.
  Fix: √(-16) is not real; it’s 4i in complex numbers.
• Mistake: Rounding too early.
  Fix: Keep a few extra decimals during intermediate steps, then round at the end.

Roots are “inverse exponents,” but they don’t behave like distribution over addition, and they come with conventions (like principal roots) that can be confusing at first. The best habit is to verify: take your computed root and raise it back to the power n.

Where roots show up outside math class

Roots are not just homework—they’re a daily tool in science, tech, and business:

• Geometry: Distance formula uses √(Δx² + Δy²) to measure straight-line distance.
• Statistics: Standard deviation includes a square root of variance.
• Physics: Many formulas solve for speed, time, or energy using square roots.
• Finance: Compound growth can be rearranged into n-th roots to solve for average growth rates.
• Computer graphics: Length normalization uses square roots to compute vector magnitudes.
• Machine learning: RMS (root mean square) error and normalization routines use roots constantly.

Suppose an investment grows from 10,000 to 12,100 in 2 years. If growth is steady, the annual growth factor is the 2nd root of 12,100/10,000: (12,100/10,000)^1/2 = 1.1, meaning about 10% per year. That “take the n-th root” step is the same operation this calculator performs.

More questions people ask about roots

• What is an “nth root” in plain English?

  It’s the number that you would multiply by itself n times to get the original number. Example: the 5th root of 32 is 2, because 2·2·2·2·2 = 32.

• Why do calculators show only one root?

  Because “the” root usually means the principal root by convention. When solving equations, you may have multiple solutions (like ±5), but calculators default to one so results are consistent and predictable.

• What if my degree isn’t an integer?

  An n-th root is typically defined with integer n. If you want something like “the 2.5th root,” you’re really asking for a fractional power: a^1/2.5. This page is designed for integer n.

• Why does 0 have only one n-th root?

  Because 0 raised to any positive power is still 0. So the only value r that satisfies rⁿ=0 is r=0.

Make this shareable (and fun)

If you’re studying with friends, here’s a quick “roots challenge” that gets shared like crazy: pick a random perfect power (like 2¹⁰ = 1024), set n, and see who can predict ⁿ√a before the calculator does. Screenshot the result card and drop it into your group chat.

• Try a = 1024, n = 5 (answer is 4)
• Try a = 59049, n = 10 (answer is 3)
• Try a = -343, n = 3 (answer is -7)