Root Calculator - Square Root, Cube Root, and nth Root

A root calculator asks for y such that yⁿ = x. On the real line that question has a unique answer when n is odd, and a non-negative answer when n is even and x ≥ 0. That distinguished value is the principal real root, written x^1/n or ⁿ√x. This page reports only that real number. It will not print −12 as “the” square root of 144, and it will not switch into complex arithmetic when you type √(−1).

The default check is the perfect square everyone knows: √144 = 12. The result panel also shows (root)ⁿ so you can see that raising 12 to the second power recovers 144. For messy radicands the recovery is approximate - floating point, not a symbolic surd.

For general powers that are not roots, use the exponent calculator or the scientific calculator. Right triangles that need √(a²+b²) belong on the Pythagorean theorem calculator.

1. Enter the radicand x. Commas are ignored.
2. Choose square (n = 2), cube (n = 3), or nth with an integer index n ≥ 2.
3. Read the principal real root and the check (root)ⁿ ≈ x. A 0th root is rejected. Even n of a negative x is an error, not a complex result.
4. Copy the value, or clear the fields. Last inputs stay in localStorage on this device for up to 30 days.

Odd roots of negatives use a real-valued path. JavaScript’s Math.pow(−8, 1/3) is NaN; this calculator still returns −2 for ∛(−8).

A square root is the n = 2 case. √144 = 12 because 12 × 12 = 144. −12 also squares to 144; the principal root keeps the non-negative one. A cube root is n = 3. ∛8 = 2 and ∛(−8) = −2. Odd powers preserve sign, so negative radicands are allowed.

An nth root generalizes both. Fourth root of 16 is 2, not −2. Fifth root of −32 is −2. n must be an integer at least 2; fractional indexes are ordinary exponents, not this tool’s job. n = 0 would ask for y⁰ = x, which is 1 for nonzero y and cannot solve for a unique y.

Any even power of a real number is ≥ 0. There is no real y with y² = −1, y⁴ = −16, or y¹⁰ = −0.5. Graphing-calculator “i” answers are complex, and they come in conjugate pairs plus extra roots of unity once n > 2. This page refuses those inputs on purpose so a homework sheet that asked for a real principal root does not get a quietly invented imaginary.

Zero is allowed: ⁿ√0 = 0 for every n ≥ 2. Tiny negative numbers from measurement noise still count as negative; the calculator will not snap them to zero.

For the principal real root, ⁿ√x and x^1/n are the same object. The identity (x^1/n)ⁿ = x holds exactly in real arithmetic when the root exists, and approximately in IEEE floating point. The residual printed here is (root)ⁿ − x. On 144 it should be 0. On √2 it should be a few units in the last place, not a new mathematical constant.

Programming languages disagree on negative bases raised to fractional powers. That is why the cube-root button exists as its own path (Math.cbrt) instead of always calling Math.pow. If you need x^y for arbitrary y, go to the exponent tool rather than forcing an nth-root index.

√2, ∛5, and the fourth root of 7 are irrational. The display is a decimal approximation with about 14 significant digits of working precision, then trimmed for reading. Perfect powers such as 144, 8, 81, and 32 are recognized when the residual is negligible, so √144 prints as 12 rather than 11.9999999997.

That rounding is for display. Copy still uses the same formatted principal root. If a teacher wants a simplified radical (2√3), this calculator will not produce that symbolic form - it is a numeric principal-root tool.

Square root and exp(ln(x)/n) are the usual library paths. Newton’s method is the picture: to solve yⁿ − x = 0, iterate y ← ((n−1)y + x / yⁿ⁻¹) / n. Starting from a rough guess, the sequence races toward the principal real root when it exists. You do not type iterations here; the residual (root)ⁿ − x is the leftover after the library has finished. A residual of 1e−12 on a number around 144 is success, not a second answer.

For huge or tiny radicands, a direct Math.pow can underflow or overflow before the root is formed. This page uses ordinary IEEE doubles, so |x| beyond about 1e308 is already not a finite input, and roots of numbers near 1e−300 may underflow to 0. Scientific notation in the input (1.44e2) is accepted by the number parser because Number() understands it.

√(16×9) = √16 × √9 = 12, which matches √144. √(a+b) is not √a + √b; √(64+81) is √145, not 8+9. Cube roots factor over products the same way, not over sums. If an assignment wants 2√3 instead of 3.464, you are in symbolic land. Numeric principal roots will not denest √(a+b±2√(ab)) into √a ± √b for you, even when that identity is available.

Nested forms such as √(6 + √(6 + √(6 + …))) have closed values for special constants, but they are limits of sequences, not a single nth-root button. Type the finite radicand you actually have. Fractional indexes like a 2/3 power are (cube root) then square, or square then cube root; use the exponent calculator if you need x^(2/3) as one operation with its own negative-base rules.

x finite, n an integer ≥ 2, and not (x < 0 and n even). That is the whole real principal-root domain this tool implements. n = 1 is excluded because it is the identity map. Non-integer n (2.5, π) is excluded because “2.5th root” is just x^(1/2.5), an exponent, and negative x would be even murkier. Complex principal roots (the branch with argument in (−π/n, π/n]) are out of scope.

If (root)ⁿ recovers x only to a few digits, suspect a very large n with x near 1, where yⁿ is extremely sensitive, or a radicand so close to a rounding boundary that the display rounded the root to an integer it should not have. Widen the residual mentally: 12.0000000002 squared is still 144 for any lab that does not live in ulps.

Roots change dimension. A length whose square is 144 m² has a principal square root of 12 m, not 12 m². A volume of 8 m³ has a cube-root side of 2 m. Putting the unit on x and then taking a root is your job; the calculator is unit-agnostic arithmetic. Area scale factors are squares of length scale factors, so a map that shrinks lengths by 2 shrinks areas by 4 - that 4 is not a root, it is the inverse story of a square.

RMS and standard deviation are square roots of means of squares. Geometric means of two positive numbers are square roots of a product; of three, a cube root of a product. Those are still principal real roots of a positive radicand, so they fit this tool once you have formed the product or the mean square. Do not cube-root a variance and call it a spread.

Inverse-square laws (light, gravity) recover distance with a square root of (constant / measured intensity) after you rearrange. Inverse-cube laws need a cube root. The algebra lives on the exponent page if you are solving xⁿ = k for arbitrary n; this page is that solve when you already know n is a small integer.

144, 81, 16, 8, and 32 are perfect powers in the defaults and worked checks. 143 is not: √143 is a little under 12, and the residual after squaring will be near zero only because of inverse rounding, not because 143 is 12². If you needed to know whether an integer is a perfect square, compare the rounded root squared to x exactly. This tool’s integer display snaps 11.999999999 to 12 when the gap is tiny relative to x, which is correct for √144 and wrong as a primality-style test of “is this a perfect power?” for arbitrary integers near a boundary.

Cube roots of integers that are not perfect cubes stay obviously non-integral (∛9 ≈ 2.08008). Use the residual as a sanity check, not as a proof of exactness in the algebraic numbers.