Square roots, cube roots and any nth root — with simplified radical form, perfect-power detection and the working shown.
A root asks which number, multiplied by itself a set number of times, gives your value. The square root of 16 is 4 because 4 × 4 = 16, and the cube root of 27 is 3 because 3 × 3 × 3 = 27. Roots are the inverse of powers, so ⁿ√x is the same as x^(1/n). Enter a number and a root index below to get the exact value, the simplified radical and every step.
Negative numbers work with odd roots — the cube root of −27 is −3. Even roots of negatives have no real answer.
A root is the inverse of a power. Where 4² = 16 raises a number, √16 = 4 works backwards to find which number was squared. The parts have names worth knowing:
When no index is written, it means a square root — √16 is shorthand for ²√16. The whole expression ⁿ√x is called a radical or a radical expression.
Every root can be written as a fractional exponent, which is why the same algebra rules cover both.
| Radical form | Exponent form | Example |
|---|---|---|
| √x | x^(1/2) | √16 = 16^0.5 = 4 |
| ∛x | x^(1/3) | ∛27 = 27^(1/3) = 3 |
| ⁴√x | x^(1/4) | ⁴√625 = 5 |
| ⁿ√(xᵐ) | x^(m/n) | 16^(3/4) = 8 |
| 1/√x | x^(−1/2) | 16^(−0.5) = 0.25 |
If you have a power and want the result, use the Exponent Calculator. If you have the result and want the power itself, use the Log Calculator. Roots, powers and logarithms are three views of one relationship.
These three operations describe one relationship from three angles. Given rⁿ = x, each one solves for a different unknown.
| Feature | Roots | Powers | Logarithms |
|---|---|---|---|
| Example | √16 = 4 | 4² = 16 | log₄(16) = 2 |
| What it finds | The value being raised | The result | The exponent |
| You know | Result and exponent | Value and exponent | Value and result |
| Question asked | What squared gives 16? | What is 4 squared? | 4 to what power gives 16? |
| Notation | ⁿ√x | xⁿ | log_b(x) |
| Calculator | This page | Exponent Calculator | Log Calculator |
Take 2³ = 8 as the worked case. The power view says 2 cubed is 8. The root view says ∛8 = 2. The logarithm view says log₂(8) = 3. Same fact, three questions.
These identities let you split, combine and simplify radicals without a calculator. They are the rules exams expect you to apply.
| Identity | Example | Note |
|---|---|---|
| √(ab) = √a × √b | √36 = √4 × √9 = 2 × 3 = 6 | How radicals are simplified |
| √(a/b) = √a ÷ √b | √(16/4) = 4 ÷ 2 = 2 | b must not be 0 |
| ⁿ√(aⁿ) = a | ³√(5³) = 5 | Root and power cancel |
| (ⁿ√a)ⁿ = a | (√7)² = 7 | The check this calculator performs |
| √(x²) = |x| | √((−5)²) = √25 = 5 | Not −5 — the result is always positive |
| ⁿ√a × ⁿ√b = ⁿ√(ab) | ∛2 × ∛4 = ∛8 = 2 | Indices must match |
| ᵐ√(ⁿ√a) = ᵐⁿ√a | √(√16) = ⁴√16 = 2 | Nested roots multiply indices |
The absolute value rule catches people out. Since √(x²) = |x| rather than x, squaring then square-rooting a negative number returns its positive counterpart: √((−5)²) = 5, not −5. It matters whenever you solve an equation by taking a square root.
These appear constantly in geometry, physics and exam questions. None can be written exactly as a fraction, so their decimals never end or repeat.
| Root | Decimal (12 s.f.) | Type | Where you meet it |
|---|---|---|---|
| √2 | 1.41421356237 | Irrational | Diagonal of a unit square; A-series paper ratio |
| √3 | 1.73205080757 | Irrational | Equilateral triangle height; three-phase power |
| √5 | 2.2360679775 | Irrational | Golden ratio, (1+√5)/2 |
| √6 | 2.44948974278 | Irrational | Products of √2 and √3 |
| √7 | 2.64575131106 | Irrational | Common exam surd |
| √8 | 2.82842712475 | Irrational | Simplifies to 2√2 |
| √10 | 3.16227766017 | Irrational | Halfway point of a log scale |
| ∛2 | 1.25992104989 | Irrational | Doubling a cube's volume |
| ∛3 | 1.44224957031 | Irrational | Scaling volumes |
| ∛5 | 1.70997594668 | Irrational | Volume calculations |
√2 has a famous history: it was the first number proved irrational, and the proof — that no fraction can equal it — reportedly unsettled the Pythagoreans, who had assumed every quantity was a ratio of whole numbers.
| Key | Action |
|---|---|
| Enter | Calculate |
| Esc | Clear everything |
| Ctrl + C (or ⌘ + C) | Copy the result, when no text is selected |
| Tab | Move between the number, root and buttons |
Enter your number, choose a root from the list, or pick "Custom nth root" for anything else. Press Calculate or the Enter key. You get the decimal value, the simplified radical form where one exists (√72 becomes 6√2), whether the number is a perfect power, and a check multiplying the answer back out. The live display above the inputs shows the radical as you type.
The square root of x is the number that, multiplied by itself, gives x. Every positive number actually has two square roots — 4 and −4 both square to 16 — but the radical sign √ refers to the principal square root, which is the positive one. That is why √16 = 4, not ±4.
When solving an equation such as x² = 16 you must include both, giving x = ±4. The distinction catches people out constantly: the square-root function returns one value, while the equation has two solutions.
The cube root of x is the number that appears three times in the product. ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots handle negatives without difficulty: ∛(−27) = −3, since multiplying three negatives keeps the sign negative.
Every real number has exactly one real cube root, positive or negative, which makes cube roots simpler than square roots in that respect.
The pattern generalises to any index. ⁴√625 = 5 because 5⁴ = 625, and ⁵√32 = 2 because 2⁵ = 32. The key rule concerns the sign:
odd index + negative number → one real answer, negative e.g. ∛(−8) = −2even index + negative number → no real answer e.g. √(−4) is undefined in real numbersany index + positive number → always a real answer
The reason is straightforward. Multiplying an even count of negatives gives a positive, so an even root can never produce a negative radicand. An odd count keeps the sign, so odd roots reach negative values comfortably.
A radical is in simplest form when the radicand has no factor that is a perfect power of the index. Pull those factors out front:
The method is to find the largest perfect square (or cube, or nth power) that divides the radicand, take its root, and leave the rest inside. This calculator does it automatically and shows the split.
| Radical | Factorised | Simplified |
|---|---|---|
| √8 | √(4 × 2) | 2√2 |
| √18 | √(9 × 2) | 3√2 |
| √32 | √(16 × 2) | 4√2 |
| √48 | √(16 × 3) | 4√3 |
| √50 | √(25 × 2) | 5√2 |
| √72 | √(36 × 2) | 6√2 |
| √98 | √(49 × 2) | 7√2 |
| ∛54 | ∛(27 × 2) | 3∛2 |
| ∛128 | ∛(64 × 2) | 4∛2 |
Simplified form is exact, while a decimal is only an approximation — 6√2 is precisely correct where 8.485 is rounded. That is why exams ask for surd form.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| n² | 1 | 4 | 9 | 16 | 25 | 36 | 49 | 64 | 81 | 100 |
| n | 11 | 12 | 13 | 14 | 15 | 16 | 17 | 18 | 19 | 20 |
| n² | 121 | 144 | 169 | 196 | 225 | 256 | 289 | 324 | 361 | 400 |
A quick test: perfect squares only ever end in 0, 1, 4, 5, 6 or 9. Any number ending in 2, 3, 7 or 8 cannot be one.
| n | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 |
|---|---|---|---|---|---|---|---|---|---|---|
| n³ | 1 | 8 | 27 | 64 | 125 | 216 | 343 | 512 | 729 | 1,000 |
| √n | Value | Type | Simplified |
|---|---|---|---|
| √1 | 1 | Exact | 1 |
| √2 | 1.41421356237 | Irrational | √2 |
| √3 | 1.73205080757 | Irrational | √3 |
| √4 | 2 | Exact | 2 |
| √5 | 2.2360679775 | Irrational | √5 |
| √8 | 2.82842712475 | Irrational | 2√2 |
| √9 | 3 | Exact | 3 |
| √10 | 3.16227766017 | Irrational | √10 |
| √16 | 4 | Exact | 4 |
| √25 | 5 | Exact | 5 |
| √50 | 7.07106781187 | Irrational | 5√2 |
| √100 | 10 | Exact | 10 |
A root is rational when it gives a whole number or a simple fraction — √16 = 4 and ∛125 = 5. It is irrational when the decimal never ends or repeats, as with √2 ≈ 1.41421356…
The Greeks discovered irrational numbers through exactly this: the diagonal of a unit square is √2, and no fraction can express it. That discovery reportedly caused a crisis among the Pythagoreans, who had believed every quantity was a ratio of whole numbers.
Pythagoras' theorem finds a diagonal or hypotenuse using a square root.
c = √(a² + b²)Squaring a room, setting out foundations and checking diagonals all rely on roots.
diagonal = √(w² + l²)Standard deviation is the square root of variance, restoring the original units.
σ = √varianceRoot mean square error measures model accuracy, and distance metrics use square roots.
RMSE = √(Σe²/n)RMS voltage — the root mean square — describes the effective value of alternating current.
V_rms = V_peak/√2Pendulum period, escape velocity and wave speed all involve square roots.
T = 2π√(L/g)Annualising volatility multiplies by the square root of the number of periods.
σ_annual = σ_daily × √252Vector length and lighting calculations use square roots constantly.
|v| = √(x²+y²+z²)1) 7 2) 12 3) 4 4) 10 5) 0
1) 7√2 2) −5 3) 5 4) 4∛2 5) Because no real number squared gives a negative result
What is a square root?
The square root of a number is the value that, multiplied by itself, gives that number. The square root of 16 is 4, because 4 times 4 equals 16.
What is a cube root?
The cube root of a number is the value that appears three times in the product. The cube root of 27 is 3, because 3 times 3 times 3 equals 27.
What is an nth root?
An nth root is the value that, raised to the power n, gives your number. The fourth root of 625 is 5, because 5 to the power 4 equals 625.
What is a radical?
A radical is a root expression written with the radical sign. The number under the sign is the radicand and the small number giving which root is the index.
What is the principal square root?
The principal square root is the positive one. Although both 4 and minus 4 square to 16, the radical sign refers only to the positive value, so the square root of 16 is 4.
Can square roots be negative?
The principal square root of a positive number is always positive, but an equation such as x squared equals 16 has two solutions, positive 4 and negative 4.
Can you take the square root of a negative number?
Not in real numbers, because no real number squared gives a negative result. In complex numbers the square root of minus 4 is 2i.
Can cube roots be negative?
Yes. The cube root of minus 27 is minus 3, because multiplying three negative numbers keeps the result negative. All odd roots handle negatives this way.
Why can odd roots handle negatives but even roots cannot?
Multiplying an even number of negatives gives a positive result, so an even root can never produce a negative radicand. An odd number of negatives keeps the sign negative, so odd roots reach negative values.
How do you simplify a radical?
Find the largest perfect power of the index that divides the radicand, take its root outside the sign, and leave the rest inside. The square root of 72 becomes 6 root 2.
What is the square root of 2?
The square root of 2 is approximately 1.41421356. It is irrational, so its decimal expansion never ends or repeats and it cannot be written as a fraction.
What is the square root of 64?
The square root of 64 is 8, because 8 times 8 equals 64. It is a perfect square.
What is the cube root of 125?
The cube root of 125 is 5, because 5 times 5 times 5 equals 125.
How do roots relate to exponents?
A root is a fractional exponent. The nth root of x is the same as x to the power one over n, so the cube root of 8 equals 8 to the power one third, which is 2.
What is a perfect square?
A perfect square is a whole number multiplied by itself, such as 1, 4, 9, 16 and 25. Its square root is always a whole number.
What is an irrational root?
An irrational root is one whose decimal never ends or repeats, such as the square root of 2. It cannot be written exactly as a fraction, so surd form is more precise than a decimal.
How do fractional roots work?
A fractional exponent combines a root and a power. Sixteen to the power three quarters means the fourth root of 16 cubed, which is 8.
How are roots used in geometry?
Pythagoras theorem uses a square root to find the hypotenuse or a diagonal. For a room 3 by 4 metres, the diagonal is the square root of 9 plus 16, which is 5 metres.
How are roots used in statistics?
Standard deviation is the square root of the variance. Taking the root converts squared units back to the units of the original data, which makes the figure readable.
How are roots used in machine learning?
Root mean square error measures how far predictions fall from the truth, and distance measures such as Euclidean distance use square roots to compare data points.
Is this Root Calculator free?
Yes. It is free with no sign-up, works on any device, and runs entirely in your browser so your entries stay private.