Calculate square roots, cube roots, and any nth root instantly, with full step-by-step work and a verification check. Handles odd and even roots of negative numbers correctly.
Enter a value and a root degree (2 for square root, 3 for cube root, etc.)
This root calculator finds the square root, cube root, or any nth root of a number instantly, with full step-by-step work and an automatic verification check. It works as a square root calculator, cube root calculator, nth root calculator, and radical calculator combined — a math root calculator online built to find the real root of a number for any positive whole-number degree n. It correctly handles the tricky sign rule that separates roots: even roots of negative numbers (like square roots) are not real, while odd roots of negative numbers (like cube roots) are real and negative.
Given a value x and a root degree n, this tool computes ⁿ√x by converting the root into the fractional exponent x^(1/n). It handles square roots (n=2), cube roots (n=3), and any higher-degree root, correctly flags an even root of a negative number as not real, and verifies every result by raising it back to the n-th power.
Algebra and geometry students solving radical expressions, physics and engineering students computing RMS values or scaling relationships, finance students calculating a compound annual growth rate, statisticians converting variance into standard deviation, and anyone who needs a fast, correct root calculation can use this tool.
A root answers the inverse question to a power: instead of asking what x raised to n equals, it asks what number, raised to n, produces x. Roots are the foundation of the Pythagorean theorem, standard deviation, and any relationship where a quantity scales with area or volume rather than length — which is why the square root and cube root specifically show up so often.
Geometry uses the square root in the Pythagorean theorem to find a diagonal or hypotenuse. Physics uses root-mean-square (RMS) calculations for AC voltage and current, and square roots in projectile motion time equations. Finance uses an nth root to compute a compound annual growth rate (CAGR) from a starting and ending value. Statistics defines standard deviation as the square root of variance. Engineering uses cube roots for volume-to-length scaling in structural models.
How square roots, cube roots, and nth roots connect to fractional exponents
Every root can be rewritten as x raised to 1/n — the same operation the Exponent Calculator handles directly.
Even roots (2, 4, 6, ...) of a negative number are not real. Odd roots (3, 5, 7, ...) of a negative number are real and negative.
Raising the computed root back to the n-th power should return the original value — the fastest way to catch an error.
From entering a value to verifying the root
Type the number you want to find the root of (the radicand) — positive, negative, or a decimal.
Type the root degree: 2 for square root, 3 for cube root, 4 for fourth root, and so on.
The calculator converts the root to a fractional exponent (x^(1/n)) and evaluates it automatically.
View the computed root value, or a message explaining that an even root of a negative number is not real.
Review the automatic check that raises the result back to the n-th power to confirm it reproduces the original value.
Multiply the result by itself n times by hand to double-check it matches your original value.
Comparing ∛27, ∛−27, and √−27 to see the even/odd root sign rule in action
Find the cube root of 27, the cube root of −27, and the square root of −27, to see how the sign of the value and the parity of the root degree interact.
Explanation: This example shows why the root degree's parity matters as much as the value's sign. ∛−27 = −3 is a valid real root because an odd number of negative multiplications stays negative, but √−27 has no real answer because any real number squared (an even number of multiplications) is always non-negative.
What the sign of the value and the root degree tell you
| Value (x) | Root Degree (n) | Result |
|---|---|---|
| Positive | Any | A single positive real root (this calculator's convention) |
| Negative | Odd (3, 5, 7, ...) | A real, negative root |
| Negative | Even (2, 4, 6, ...) | Not a real number (complex) |
| Zero | Any | 0 (the n-th root of 0 is always 0) |
Reading the result: a positive result means the value and root degree combination is straightforward; a negative result confirms an odd-degree root of a negative number; "not a real number" confirms an even-degree root of a negative number, which this calculator flags rather than returning an incorrect value.
Typical/edge cases: the root of 0 is always 0, and the root of 1 is always 1, regardless of the root degree. Very large values can produce roots with small decimal remainders even when the input looks like a "clean" number.
Manual verification: multiply the result by itself n times and confirm it reproduces the original value — this calculator does this automatically and displays it as the verification line.
Because this calculator follows the principal-root convention, returning the single non-negative value that, raised to the n-th power, reproduces x. Mathematically, 9 has both +3 and −3 as square roots, but for a positive value the principal n-th root is the positive one, so that is the answer shown here.
A decimal radicand works exactly like a whole number — the tool treats 2.25^(1/2) the same as √2.25, which equals 1.5. The only difference is that many decimals produce non-terminating results, such as √2 ≈ 1.4142136, so the answer is displayed with its trailing zeros trimmed.
It almost always means floating-point rounding rather than an error. Raising a computed root back to the n-th power can leave a tiny trailing difference, like 26.9999999 instead of 27, especially for large values or high degrees. If the verification rounds back to your original value, the root is correct.
If the value is a perfect power for that degree — 8 as a cube root, 81 as a fourth root — the result is a clean whole number and therefore rational. If not, the result is irrational and appears as a decimal that never terminates, such as √2 ≈ 1.4142136, which no finite decimal can capture exactly.
The same sign rule applies. Because the cube root degree 3 is odd, ∛(−27/8) is a real negative number: −(3/2), or −1.5, since (−1.5) × (−1.5) × (−1.5) = −3.375 = −27/8. An even root of that same negative fraction would instead be reported as not a real number.
Large radicands push the computation deep into floating-point territory, so the result can carry extra digits after the decimal even when the input looks round. If you prefer a compact answer, pass the same result through the Scientific Notation Calculator to express it as a × 10ᵇ.
Where finding a square, cube, or nth root genuinely comes up
Check square root, cube root, and radical simplification problems step by step.
Find a triangle's hypotenuse or diagonal length, which always requires a square root.
Convert a computed variance into standard deviation by taking its square root.
Compute root-mean-square values for AC electrical signals using a square root.
Find an investment's annualized growth rate, which involves taking an nth root.
Use cube roots for volume-to-length relationships when scaling models or structures.
Compute Euclidean distance between two points, which requires a square root.
Solve formulas involving root-mean-square error or root-based scaling laws.
Use square roots in complexity analysis (e.g. O(√n) algorithms) and hashing functions.
Instructors verify root and radical problems and generate step-by-step explanations.
Find diagonal bracing lengths or square footage side lengths using roots.
Builds the fractional-exponent intuition needed before using the Exponent Calculator.
Anyone who needs a fast, reliable square root, cube root, or nth root answer.
What this root calculator does well, and where care is still needed
How root behavior changes with the degree n
| Root Type | Degree (n) | Real Root of a Negative Number? | Example |
|---|---|---|---|
| Square root | 2 (even) | No — not real | √−16 → not real |
| Cube root | 3 (odd) | Yes — real and negative | ∛−8 = −2 |
| Fourth root | 4 (even) | No — not real | ⁴√−16 → not real |
| Fifth root | 5 (odd) | Yes — real and negative | ⁵√−32 = −2 |
Summary: This root calculator gives you an instant, free way to find the square root, cube root, or any nth root of a number, correctly handling negative values based on whether the root degree is odd or even, with a verification check on every answer. Pair it with related tools like the Exponent Calculator and Scientific Notation Calculator for a fuller picture of powers, roots, and scale.
Common questions about square roots, cube roots, and nth roots
Trusted educational references to go deeper on roots
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