A perfect square
- Square root
- 12
- Radical form
- 12
- Verification
- (12)^2 ≈ 144
When the input is exact, all three fields agree and nothing is rounded. This is the case that makes the other cases readable.
Math
Square Root Calculator — Nth Roots
Calculate square, cube and integer nth roots, simplify square radicals, and check the result. Supports negative numbers with odd root degrees.
Formula and sources updated · Data sources · 1View algorithm, constants and default inputs (JSON)
Calculation steps
- √144 = 12
- Nth root ≈ 12
- Verification: (12)^2 ≈ 144
About this calculator
Enter a number and choose the root degree: 2 for a square root, 3 for a cube root, or another integer up to 100. The selected degree controls the main answer. Negative numbers need an odd degree. Nonnegative integers equal to or below 1,000,000,000,000 also show a simplified square root.
Recomputed from this page's implementation at build time · checked 2026-10-22
The radical form always simplifies the square root, even when you ask for another root
Three numbers describe the same input here: a decimal approximation, an exact radical form, and a verification line that restates the definition. They answer to different rules, and one of them ignores the root selector entirely.
- 12 The square root of 144 exact, and printed without decimals
- 1.414213562373 The square root of 2 at twelve decimals the precision comes from the input, not the page
- 10√2 200 pulled apart the largest square factor extracted
Where the panel goes blank
-
You typed
- value
- -9
- root
- 2
- precision
- 6
It answeredrealRootErrorA negative input with an even root
The cube root still reports -2.080084, and the squared value is 81.
No real number squares to a negative value, so the square root is unavailable. The panel does not blank out everything: the odd root and the square remain valid and keep their numbers, which is why an error message and live decimals appear on the same screen.
-
You typed
- value
- 10000000000000000
- root
- 2
- precision
- 6
It answeredinputErrorOne step past the accepted range
The square root field still shows 100,000,000 while the n-th root is empty.
The guard is on the input size, and it fires after some fields have already been computed. The result is a partly populated panel rather than a clean refusal — worth knowing before you trust a single number from it.
How the result moves
| 2 | 10 | 100 | 10000 | 1000000 | |
|---|---|---|---|---|---|
| 2 | 1.414 | 3.162 | 10.000 | 100.000 | 1000.000 |
| 3 | 1.260 | 2.154 | 4.642 | 21.544 | 100.000 |
| 4 | 1.189 | 1.778 | 3.162 | 10.000 | 31.623 |
A million is worth a thousand as a square root, a hundred as a cube root, and about thirty-two as a fourth root. The higher the degree, the more the operation behaves like a constant rather than a scaling.
Three fields, three rules
A non-square with one visible square factor
- Square root
- 14.142136
- Radical form
- 10√2
- Verification
- (14.142135623731)^2 ≈ 200
The radical form is exact and the decimal is not. If you have to carry the value into further algebra, the radical survives it; the decimal accumulates error from the first multiplication.
The same input read as a third root
- N-th root
- 3
- Square root
- 5.196152
- Radical form
- 3√3
- Verification
- (3)^3 ≈ 27
Two different roots appear on one screen. The third root of 27 is exactly 3, while the radical field still simplifies the square root of 27. That field is bound to the square root and does not follow the selector.
Two, at twelve decimals
- Square root
- 1.414213562373
- Radical form
- √2
- Verification
- (1.4142135623731)^2 ≈ 2
More digits do not make the decimal exact, and they never touch the radical. Precision is a display setting for one field, not a change to the calculation.
A negative input with an odd root
- N-th root
- -3
- Cube root
- -3
- Verification
- (-3)^3 ≈ -27
This line computes. An odd root of a negative number is a real number, so refusing every negative input would be wrong — the next section shows the case that actually fails.
Which field obeys what
- The radical field belongs to the square root
- Simplification factors the input into a square part and a remainder, then writes the square part outside the sign. It never consults the root selector. When the selector is set to 3 or 4, the radical shown still describes a square root.
- Precision is a display control
- It sets how many decimals are printed for the root fields. Twelve is the maximum and does not imply twelve correct digits for irrational inputs — it implies twelve printed ones.
- The verification line restates the definition
- It prints the computed root raised back to the selected power, next to the original input. That is the cheapest check on the whole page: if the two sides disagree visibly, the rounding is too coarse for your purpose.
- At the top of the range the radical form gives up
- For very large accepted inputs the radical field returns nothing while the decimal still computes. An empty field here means the page declined to simplify, not that the value has no root.
How this compares
-
Calculator.net root calculatorchecked 2026-10-22
- Theirs
- Three separate boxes on one page — square root, cube root, and a general n-th root — each printing a rounded answer, with no radical simplification and no precision control.
- Here
- One input and a root selector, a decimal whose precision you choose, an exact radical form, and a verification line that multiplies the result back.
Why they built it that waySquare roots and n-th roots sit on separate pages with separate forms. The same reasoning as the statistics set: the two phrases are typed as different questions, and a page that answered both would be competing with itself.
Simplification is what their three boxes cannot do: 200 comes out as 14.142 there and as 10√2 here, and only one of those survives being squared again by hand. The trade is on this page's side — its radical field is wired to the square root, so at a root of three it answers a different question from the decimal beside it.
Enter 27 and set the root to 3: the n-th root reads 3 while the radical field still simplifies the square root. Then enter -9 and confirm the square root is unavailable while the cube root keeps a number.
Compared against the closed-form power, and against the cube root of a perfect cube · impl 0da95403
Method and assumptions
The square-root result is the nonnegative principal square root: for example, √9 = 3. For a selected degree n, the main result y satisfies yⁿ = x; negative x is supported only for odd n. To simplify a square radical, factor out perfect squares: √72 = √(36 × 2) = 6√2. The separate simplified square-root output always refers to degree 2, even when another degree is selected. Decimal answers use floating-point arithmetic. Raising the rounded root to its degree gives an approximate verification, not a proof of exact equality.
Data sources
Disclaimer
These calculators give general mathematical results. They suit study, planning and estimation — verify anything that feeds a graded submission, a published figure or a financial commitment.
Implementation source
This record describes the implementation used by this page. It is not independent validation. External references, when available, support only the stated formula or definition; estimates and custom rules retain their own assumptions.
Formula and variables
y = √x (x ≥ 0); yₙ = sgn(x)·|x|^(1÷n); n ∈ ℤ, 2 ≤ n ≤ 100; x < 0 ⇒ n ∈ {3,5,…,99}
- x
- Input number
- n
- Integer root degree, from 2 to 100
- y
- Nonnegative principal square root
- yₙ
- Real nth root for the selected degree
The √ symbol gives the nonnegative principal square root. Negative inputs require an odd root degree for a real answer. The separate simplified radical always describes the square root, and is available for nonnegative integer inputs up to 1,000,000,000,000. Decimal roots and the verification are approximate.
Frequently asked questions
Can I take a root of a negative number?
Yes, if the root degree is odd. For example, the cube root of -8 is -2 because (-2)³ = -8. An even root of a negative number has no real result, so this calculator shows an error; complex roots are not supported.
What do precision and the simplified square root mean?
Decimal places controls the numeric display from 0 to 12 places; it does not change the input or provide arbitrary precision. Very small nonzero results use scientific notation. For nonnegative integer inputs up to 1,000,000,000,000, the tool also shows an exact simplified square radical such as √72 = 6√2. That separate output remains a square root when you choose another degree. Verification uses a rounded root and is approximate.
How do I simplify the square root of 200?
Choose root degree 2 and enter 200. Since 200 = 100 × 2, √200 = 10√2. The tool shows the decimal value and simplified square radical for nonnegative integers up to 10¹². A negative input needs an odd degree for a real root.
What numbers and root degrees can I enter?
Enter a number from -1,000,000,000,000,000 to 1,000,000,000,000,000. Decimals are allowed for the number, but the root degree must be an integer from 2 to 100 and Decimal places must be an integer from 0 to 12. A negative number requires an odd degree. Simplified square radicals are available only for nonnegative integer inputs up to 1,000,000,000,000.
What is the square root of 144?
The principal square root of 144 is 12, since 12 × 12 = 144. Enter 144 and choose degree 2 to see 12 as the main result and simplified square root. Although -12 also squares to 144, the √ symbol denotes the nonnegative principal square root.