Quick Answer
A square root calculator with steps finds the number that, multiplied by itself, equals your input — written √x, or the nth root ⁿ√x for higher degrees. This tool returns the decimal value, the simplified radical form (e.g. √72 = 6√2), and the full prime-factorization working.
Square Root Formula and Simplified Radical Form
A square root asks what number, multiplied by itself, gives the radicand. The calculator answers that twice — once as a decimal and once as an exact simplified radical — and it reaches the exact form by prime factorisation rather than by searching for a perfect square that happens to divide.
Write the radicand as a product of primes, n = p₁e₁ × p₂e₂ × …. For the kth root, each prime contributes p⌊e∕k⌋ outside the radical and pe mod k inside it. The result is outside × k√inside.
That one rule covers every case the tool handles. At k = 2 it simplifies square roots; at k = 3 it simplifies cube roots; at k = 20 it still works, because nothing in the rule mentions squares. The floor division decides how many copies of a prime escape the radical, and the remainder decides how many stay behind.
How to Use the Square Root Calculator
- Type a number into the radicand box.
This is the number under the root sign. Whole numbers, decimals and negatives are all accepted, and the results update as you type — there is no calculate button.
- Leave the degree on Square (√) for an ordinary square root.
This is the default, so for a plain square root you only need step 1.
- Press Cube (∛) or Fourth (∜) for those roots.
The symbol in front of the input box changes to match, so you can see which root you are taking.
- For any other root, type the index into the “Custom n” box.
Anything from 2 to 20 works. A value here overrides the buttons, so type 5 for a fifth root. Clearing the box hands control back to the buttons.
- Read the four result cards.
Decimal approximation, simplified radical form, complex form when there is no real answer, and the step-by-step working that shows the prime factorisation.
Everything runs in your browser. The number you type is never sent anywhere.
Worked Examples: Square, Cube and Nth Roots
Every figure below is what the calculator prints, at its own ten-decimal-place rounding.
| Input | Prime factorisation | Simplified form | Decimal |
|---|---|---|---|
| √72 | 2³ × 3² | 6√2 | 8.4852813742 |
| √50 | 2 × 5² | 5√2 | 7.0710678119 |
| √200 | 2³ × 5² | 10√2 | 14.1421356237 |
| √288 | 2⁵ × 3² | 12√2 | 16.9705627485 |
| √2352 | 2⁴ × 3 × 7² | 28√3 | 48.4974226119 |
| √144 | 2⁴ × 3² | 12 — perfect square | 12 |
| √2 | 2 | √2 — already simplest | 1.4142135624 |
| ∛54 | 2 × 3³ | 3∛2 | 3.7797631497 |
| ∛16 | 2⁴ | 2∛2 | 2.5198420998 |
| ∛216 | 2³ × 3³ | 6 — perfect cube | 6 |
| ⁵√96 | 2⁵ × 3 | 2 ⁵√3 | 2.4914618792 |
| ⁵√1024 | 2¹⁰ | 4 — perfect 5th power | 4 |
Read the √72 row against the ∛54 row and the mechanism is visible. In 72 = 2³ × 3², the exponent 3 divided by index 2 leaves one 2 outside and one 2 inside, while 3² sends the 3 out cleanly — so 6 comes out and 2 stays. In 54 = 2 × 3³, the same division against index 3 sends the whole 3³ out as a single 3 and strands the lone 2. Nothing about the procedure changed; only k did.
Why This Calculator Factorises Instead of Guessing
Most step-by-step square root guides teach a different method: look for the largest perfect square that divides your number, then split the radical. For √72 you are told to spot 36, giving √36 × √2 = 6√2. It gets the right answer, and for small textbook numbers it is quick.
It is also a search, not a procedure. There is no rule that tells you 36 is the square to use — you have to recognise it. Prime factorisation has no such step: the exponents tell you what comes out, and by the fundamental theorem of arithmetic every integer above 1 has exactly one prime factorisation, so the method cannot produce two different answers or leave you wondering whether you finished.
The difference shows up in two measurable ways. Taking every whole number from 2 to 1000 — 999 radicands in all:
| Measure, radicands 2–1000 | Count | Share |
|---|---|---|
| Simplify under a square root (a square factor above 1) | 392 | 39.2% |
| Of those, ones with two or more distinct square factors | 112 | 28.6% of 392 |
| Simplify under a cube root (a cube factor above 1) | 167 | 16.7% |
| Perfect squares | 30 | 3.0% |
| Perfect cubes | 9 | 0.9% |
That second row is the practical problem. In 112 of the 392 cases — more than a quarter — the radicand has several perfect-square factors, so the trick offers a wrong turn: pick a square that is not the largest, and you stop early with an answer that still simplifies.
√288 is the clean illustration. Its perfect-square factors are 4, 9, 16, 36 and 144. Spot the obvious 4 first and you get 2√72 — but the calculator shows √72 = 6√2, so 2 × 6 = 12 and the real answer was 12√2 all along. Prime factorisation reaches 12√2 in one pass, because 288 = 2⁵ × 3² and the exponents 5 and 2 already say what escapes.
The widest gap in the 2–1000 range is 900. The first perfect square you would spot dividing it is 4; the largest is 900 itself. Working from the small end costs you several rounds of simplifying to reach what one factorisation gives immediately.
Larger numbers make the search harder rather than merely longer. √2352 simplifies to 28√3, which means its largest perfect-square factor is 784 — and 784 is not a number anyone spots. Its factorisation, 2⁴ × 3 × 7², is routine.
Square Root of N in Simplified Form: Reference Table
The radicands people look up most often, with the exact form the calculator returns for each.
| N | √N simplified | Decimal | N | √N simplified | Decimal |
|---|---|---|---|---|---|
| 8 | 2√2 | 2.8284271247 | 75 | 5√3 | 8.6602540378 |
| 12 | 2√3 | 3.4641016151 | 80 | 4√5 | 8.94427191 |
| 18 | 3√2 | 4.2426406871 | 98 | 7√2 | 9.8994949366 |
| 20 | 2√5 | 4.472135955 | 108 | 6√3 | 10.3923048454 |
| 24 | 2√6 | 4.8989794856 | 128 | 8√2 | 11.313708499 |
| 27 | 3√3 | 5.1961524227 | 147 | 7√3 | 12.124355653 |
| 32 | 4√2 | 5.6568542495 | 180 | 6√5 | 13.416407865 |
| 40 | 2√10 | 6.3245553203 | 200 | 10√2 | 14.1421356237 |
| 45 | 3√5 | 6.7082039325 | 288 | 12√2 | 16.9705627485 |
| 48 | 4√3 | 6.9282032303 | 300 | 10√3 | 17.3205080757 |
| 50 | 5√2 | 7.0710678119 | 72 | 6√2 | 8.4852813742 |
Note 24 and 40. Neither ends in a single prime: 24 = 2³ × 3 leaves 2 × 3 = 6 inside, and 40 = 2³ × 5 leaves 2 × 5 = 10. The leftovers multiply together rather than staying separate, which is why the answers are 2√6 and 2√10 and not two radicals side by side.
Simplifying Surds
In British and Commonwealth syllabuses an irrational root left in exact form is called a surd. Simplifying a surd and simplifying a radical are the same operation under two names, so every row of the table above is a surd in simplified form: √50 = 5√2 is the surd 5√2.
The distinction that matters for exam work is exact versus approximate. 5√2 is exact and 7.0710678119 is not — the decimal is truncated, because the result is irrational and its expansion never terminates or repeats. The calculator labels this on the decimal card, and marks the whole-number cases separately as a perfect square, perfect cube or perfect nth power. When a question says “leave your answer in surd form”, the simplified radical card is the answer and the decimal card is not.
Cube Roots and Nth Roots
This is where the two methods separate for good. “Find the largest perfect square factor” has nothing to say about ∛54, because squares are irrelevant to a cube root — you would need the largest perfect cube factor, and then the largest perfect fifth power, and so on for every index. Each is a different search.
The factorisation rule does not change at all. 54 = 2 × 3³: divide each exponent by 3, and the 3³ contributes one 3 outside while the lone 2 stays inside, giving 3∛2. Set the index to 5 and 96 = 2⁵ × 3 gives 2 ⁵√3 by the identical step.
| Radicand | Index 2 | Index 3 | Index 7 |
|---|---|---|---|
| 128 = 2⁷ | 8√2 | 4∛2 | 2 |
One radicand, three indices, three different answers — and 128 is a perfect seventh power while being neither a perfect square nor a perfect cube. The index is not decoration on the radical sign; it is the divisor in the rule. That is also why the calculator names the case correctly: 216 under a cube root is reported as a perfect cube, and 1024 under a fifth root as a perfect fifth power, rather than calling everything a perfect square.
Cube roots also behave differently on negatives, which is the next section.
Negative Radicands: Imaginary and Complex Roots
An odd index over a negative radicand has a real answer: ∛−27 = −3 and ∛−8 = −2, because a negative number cubed is negative. The calculator keeps the decimal card in this case and prints −3.
An even index does not. √−16 has no real value, so the tool hides the decimal and simplified cards and shows the complex form 4i instead. The radical survives the move when it needs to: √−50 comes out as 5√2 i, exactly the simplified form of √50 with i attached.
Multiplying the real root by i only works at index 2. The fourth root of −16 is not 2i — check it: (2i)⁴ = +16, the wrong sign. The calculator returns 1.4142135624 + 1.4142135624i, a principal root sitting at 45° in the complex plane with a non-zero real part.
The rule behind that is the principal kth root of a negative real, which sits at an angle of 180°∕k. At k = 2 the angle is 90°, the real part is zero and the answer collapses to the familiar multiple of i. At k = 4 the angle is 45°, and both parts are non-zero. Negative discriminants under a square root are the everyday version of this — the quadratic formula calculator reports the same imaginary form when b² − 4ac is negative.
Perfect Squares, Zero and One
√0 = 0 and the calculator reports it as a perfect square, which is correct: 0 × 0 = 0. It is the one radicand with a single root rather than a positive and a negative pair, and it works at every index, so ∛0 = 0 as well. √1 = 1 for the same reason.
When the radicand is a perfect kth power the inside of the radical becomes 1, and rather than print √1 the tool drops the radical entirely and labels the result — perfect square for 144, perfect cube for 216, perfect fifth power for 1024. Recognising a perfect square matters beyond this page: in a diamond problem, the two factors are whole numbers only when S² − 4P is a perfect square.
Limits of the Calculator
Two limits are worth knowing before you trust a result, and both are visible in the interface rather than hidden.
Decimal radicands are not simplified. Type 2.25 and you get 1.5 with a note saying only the decimal approximation is shown. Simplification is defined on integer factorisations, so the simplified card hides itself rather than guessing. The decimal is still exact where it can be: √2.25 really is 1.5, and √6.25 is 2.5.
Very large radicands can defeat the factoriser. The prime search tests divisors up to 1,000,000, which covers any radicand up to about 10¹² fully. Beyond that a radicand whose smallest prime factor exceeds a million is reported as “already in simplest form” even when it is not. √1,000,006,000,009 is exactly 1,000,003 squared: the decimal card prints 1,000,003 correctly, while the simplified card leaves the radical intact. The same shape below the limit, √994,009, resolves cleanly to 997.
When the two cards disagree like that, the decimal card is the one to believe. It is computed directly and never depends on the factorisation.
Frequently Asked Questions
Yes. √0 = 0, and the calculator labels it a perfect square because 0 × 0 = 0. It is the only radicand with exactly one root instead of a positive and negative pair, and it gives 0 at every index.
Squaring. Squaring and taking a square root undo each other, so √72 = 6√2 can be checked by squaring: 6² × 2 = 72. More generally the opposite of a kth root is raising to the power k.
Write the radicand as a product of primes, then move each prime out once for every two copies of it. In 72 = 2³ × 3², the three 2s give one 2 outside with one left inside and the two 3s give a 3 outside, so √72 = 6√2.
Simplifying a radical works on the prime factorisation of an integer, and a decimal has no such factorisation. The tool hides the card and keeps the decimal approximation, which stays exact where the answer is exact — √2.25 is 1.5.
Because (2i)⁴ = +16, not −16. Attaching i to the real root only works at index 2; above it the principal root sits at 180°∕k in the complex plane, so the fourth root of −16 is 1.4142135624 + 1.4142135624i.
An odd number of negative factors stays negative, so (−3)³ = −27 and ∛−27 = −3. Even indices cannot do this, which is why √−16 has no real value while ∛−27 does.
6√2 is exact; 8.4852813742 is rounded to ten decimal places. √72 is irrational, so its decimal never terminates — give the simplified radical whenever a question asks for an exact or surd-form answer.
Yes — a surd is an irrational root left in exact form, so the simplified radical card is the surd form. Enter 50 and it returns 5√2; enter 288 and it returns 12√2.
The prime search stops at divisors of 1,000,000, so a radicand whose smallest prime factor is larger goes unfactored. √1,000,006,000,009 is 1,000,003 squared and the decimal card prints 1,000,003 correctly — trust the decimal card when the two disagree.
