What is the n-th root?
The n-th root of a number undoes raising that number to the n-th power. Square root (n=2) undoes squaring. Cube root (n=3) undoes cubing. The fourth root undoes raising to the fourth power. The pattern keeps going for any positive integer n. Written with a radical sign, the n-th root of x looks like ⁿ√x. Written with an exponent, it's x^(1/n). They mean the exact same thing — the radical is older notation, the exponent form is more flexible.
The intuition is reversal. If 2⁵ = 32, then ⁵√32 = 2. If 4³ = 64, then ³√64 = 4. You are asking the question: "what number, raised to the n-th power, gives me this?" Sometimes the answer is a clean integer (those numbers are called perfect n-th powers). Most of the time the answer is irrational and you read off a decimal approximation.
Roots show up in geometry (cube root of a volume gives a side length), finance (compound annual growth rate is a root operation), physics (acceleration formulas involve square roots), statistics (standard deviation is a square root), and audio (the RMS in "RMS volume" is literally a root mean square). Anywhere you need to walk back a power, you reach for a root.
How to use the Root Calculator
The Root Calculator takes two inputs: the radicand (the number you want to take the root of) and the index n (which root). It returns the result instantly. There is no Calculate button.
- Enter the radicand. Positive numbers always work. Negative numbers work for odd indices only (more on this below).
- Enter the root index n. Use 2 for square root, 3 for cube root, 4 for fourth root, and so on. Any positive integer is fine.
- Read the result. If the radicand is a perfect n-th power, you get an exact integer answer with a confirmation like "Exact integer: 4³ = 64." Otherwise you get a decimal to 10 places.
- Underneath, the calculator shows the equivalent exponent form: ⁿ√x = x^(1/n). That is the actual identity it uses to compute the answer.
Inputs stay in your browser. Nothing is sent anywhere. There is no sign-up, no quota, no trial countdown — most root calculators online either gate the result behind a newsletter or open with a banner ad larger than the answer. This one just returns the number.
Worked examples
Two clean cases, each chosen because the answer is a perfect integer:
The cube root of 27. You are asking: what number, cubed, equals 27? Test 2: 2³ = 8, too small. Test 3: 3³ = 27. So ³√27 = 3 exactly. The calculator detects this and returns 3 with the exact-integer label.
The fourth root of 16. You are asking: what number, raised to the fourth power, equals 16? Test 2: 2⁴ = 2·2·2·2 = 16. So ⁴√16 = 2 exactly. Again the calculator flags it as an exact integer.
A less clean case: the cube root of 50. Test 3: 27, too small. Test 4: 64, too big. The answer lives between 3 and 4. The calculator returns 3.6840314986… as a decimal, with no exact-integer flag — 50 is not a perfect cube. Most real-world inputs land in this irrational bucket. Perfect n-th powers are rare; the integers 1, 4, 9, 16, 25, 36, 49 are perfect squares, and the gaps between perfect cubes (1, 8, 27, 64, 125) get wider fast.
The defining identity: ⁿ√x = x^(1/n). The square root of 9 is 9^(1/2) = 3. The cube root of 8 is 8^(1/3) = 2. The Root Calculator computes everything via this single rule, using JavaScript's Math.pow under the hood. Same identity, no special cases for different indices.
Negative radicands and why even roots refuse them
Take any real number and square it. The result is positive (or zero). 3² = 9. (-3)² = 9. 0² = 0. There is no real number whose square is negative because two negatives multiplied give a positive. So the square root of -4 has no real answer — it is the imaginary number 2i, which lives off the real number line.
The same logic blocks every even root. Fourth root of -16? No real answer. Sixth root of -1? No real answer. Any even power of a real number is non-negative, so any even root of a negative number is imaginary.
Odd roots have no such problem. The cube root of -8 is -2, because (-2)³ = -2 · -2 · -2 = -8. Three negatives multiplied still give a negative. The same holds for every odd index: fifth root of -32 is -2, seventh root of -128 is -2, and so on. If the radicand is negative and n is odd, the result is negative real. If the radicand is negative and n is even, the Root Calculator returns an error rather than silently producing nonsense.
If you actually need complex-number roots (electrical engineering, signal processing, quantum mechanics), you need a complex-number calculator. That is a deliberate scope choice — the math gets meaningfully more involved and the audience for it is small.
Comparing different roots of the same number
Watching how the result changes as n grows is one of the more satisfying things about roots. Take 64 and step through:
| Index n | Expression | Result | Exact? |
|---|---|---|---|
| 2 | ²√64 | 8 | Yes (8² = 64) |
| 3 | ³√64 | 4 | Yes (4³ = 64) |
| 4 | ⁴√64 | 2.8284271247… | No |
| 5 | ⁵√64 | 2.2973967099… | No |
| 6 | ⁶√64 | 2 | Yes (2⁶ = 64) |
| 10 | ¹⁰√64 | 1.5157165665… | No |
| 100 | ¹⁰⁰√64 | 1.0425103998… | No |
Three perfect cases (n = 2, 3, and 6) and a clear trend: as n grows, the result heads toward 1. That makes sense — the larger n is, the smaller the slice of the original "power" you are walking back. For n = 1000, ¹⁰⁰⁰√64 ≈ 1.00417. As n approaches infinity, every positive radicand greater than 1 has its n-th root approach 1 from above.
For radicands between 0 and 1, the opposite happens — the n-th root approaches 1 from below. The fifth root of 0.5 is 0.8706. The hundredth root of 0.5 is 0.9931. The thousandth root of 0.5 is 0.9993. Again, the n-th root of any positive number tends to 1 as n grows.
Where roots show up in practice
A few places worth knowing:
- Geometry — side from volume. A cube with volume 1000 cm³ has a side length of ³√1000 = 10 cm. Same idea for any n-dimensional hypercube.
- Compound growth — CAGR. If an investment grows from $1,000 to $1,610.51 over 5 years, the compound annual growth rate is ⁵√(1610.51/1000) − 1 = 0.10, or 10% per year. The n-th root undoes the compounding.
- Audio — RMS levels. The "RMS" in "RMS volume" stands for root mean square: square each sample, average them, take the square root. The square root step is what gives the result its sensible dimensions (the same as the input signal).
- Statistics — standard deviation. Variance is the mean of the squared deviations. Standard deviation is the square root of variance, which restores the original units. A salary variance has units of "dollars squared," which is meaningless; a standard deviation has units of dollars.
- Physics — free fall. Time to fall distance d in Earth gravity is √(2d/g). Drop something from 20 meters and it hits the ground in √(40/9.8) ≈ 2.02 seconds.
Most calculators bundle these into separate apps (CAGR calculator, free-fall calculator, audio RMS calculator). The math underneath is the same root operation. If you understand the n-th root, you have already done 80% of the work on every one of those formulas.
How the calculator detects exact integer roots
Floating-point arithmetic is imprecise. Math.pow(64, 1/3) does not return exactly 4 — it returns something like 3.9999999999999996 because 1/3 is not representable exactly in binary. A naive calculator would print that ugly decimal and call it a day.
The Root Calculator does an extra step: round the result to the nearest integer, then raise that integer back to the n-th power and check whether you recover the original radicand exactly. If yes, the answer is a perfect n-th power and the calculator displays the clean integer. If no, the result is genuinely irrational and you get the decimal.
The check is cheap (one extra multiplication) and catches the common case where you typed a perfect cube and would rather see "4" than "3.9999999999999996." Same logic for square roots, fourth roots, and beyond. If you need precision higher than the 15-17 significant digits of double-precision floats, you would need an arbitrary-precision library — but for almost every practical purpose, the calculator's output is plenty.
Related calculators
The Root Calculator is the general-purpose tool. For specific cases or related operations, these are usually faster:
- Square Root Calculator — dedicated to the n=2 case. Same math, simpler interface if you only ever need square roots.
- Exponent Calculator — the inverse direction. Raise a number to any power. Useful when you want to verify a root result by raising it back.
- Pythagorean Theorem Calculator — the most common everyday use of square roots: finding the hypotenuse of a right triangle from its two legs.
Frequently asked questions
What is the formula?
The n-th root of x equals x to the 1/n power: ⁿ√x = x^(1/n). The square root of 9 is 9^(1/2) = 3. The cube root of 27 is 27^(1/3) = 3. The calculator uses JavaScript's Math.pow with the appropriate fractional exponent, then does the integer-check pass described above.
Why can't I take the square root of a negative number?
Because no real number, when squared, gives a negative result. Any real number times itself is positive or zero. The square root of -4 is the imaginary number 2i, which is not on the real number line. The Root Calculator handles real numbers only. If you need complex-number roots, you need a different tool.
But I can take the cube root of a negative number?
Yes — for odd indices (cube root, fifth root, seventh root, and so on) of negative numbers, the result is a negative real number. The cube root of -8 is -2 because (-2)³ = -8. The pattern holds for all odd indices. For even indices of negative numbers, the result is complex and the calculator returns an error.
How does the calculator detect exact integer roots?
It computes the root via Math.pow, rounds to the nearest integer, and checks whether that integer raised to the original power gives back the input exactly. If yes, perfect n-th power, display the integer. If no, the result is irrational and the decimal stands. Example: cube root of 64 produces 4.000000000001 from floating point, round to 4, check 4³ = 64, confirmed, display 4.
Why are some decimals slightly off?
JavaScript numbers are 64-bit IEEE 754 doubles with roughly 15-17 significant digits. For irrational roots like √2 = 1.41421356237…, the calculator returns the best double-precision approximation. For perfect roots the integer-check pass corrects the rounding. If you genuinely need more than 15 digits of precision, you need an arbitrary-precision math library — but for almost all practical uses, double precision is well beyond what you can verify by hand.
How do I write a fourth root or higher?
On paper, the index sits as a small superscript on the radical: ⁴√x. In code or plain text, "the 4th root of x" or "x^(1/4)" works fine — they all mean the same thing. The Root Calculator accepts any positive integer for n. Larger n means the result is closer to 1 for radicands above 1, and closer to 1 from below for radicands between 0 and 1.
Is the square root and the 1/2 power the same thing?
Yes, by definition. √x = x^(1/2). The radical notation is older and more common in schoolwork; the exponent notation is more flexible because you can write x^(2/3) for the cube root of x squared, which is awkward to draw with a radical. Either form works.
Why does the root of a small number give a result close to 1?
Because of how exponents behave near zero. Any positive number raised to a small fractional power approaches 1 — even 1,000,000^(1/100) ≈ 1.148. Intuition: the n-th root walks back n self-multiplications, and for large n that walk-back is mild. For radicands above 1, the root shrinks toward 1 as n grows. For radicands between 0 and 1, the root grows toward 1 as n grows. Either way, big n flattens the answer.