This calculator finds either the principal square root or the real cube root of a number. A root reverses a power: a square root reverses squaring, and a cube root reverses cubing. The two operations have different real-number domains. The principal square root requires a nonnegative input, while a cube root exists for positive, zero and negative real numbers.
Root Calculator
Calculate either the principal square root of a nonnegative number or the real cube root of any real number.
Calculator inputs
Square or cube root
Enter your values
Calculated output
Result summary
Your result will appear here
Complete the required fields and select the calculate button.
Calculated root√x = x¹ᐟ²; ∛x is the real number whose cube is x
- Calculated root
- —
- Root type
- —
- Entered value
- —
- Power check
- —
The square-root branch returns the nonnegative principal root and does not accept negative real inputs. The real cube-root branch accepts negative values.
What is Root Calculator?
How to Use This Calculator
- Choose Square Root or Cube Root from the menu. The form displays only the input that belongs to that choice.
- For a square root, enter zero or a positive number. For a cube root, enter any real number supported by the numeric field.
- Select Calculate. The page evaluates the chosen root and rounds the displayed result to four decimal places.
You can check a square-root result by squaring it and a cube-root result by cubing it. A rounded answer may not return the input exactly when raised back to the power, so use the unrounded value for later calculations when precision matters.
Formula Used
Principal square root: √x = x1/2, for x ≥ 0 in the real numbers.
Real cube root: ∛x = x1/3.
If y = √x, then y ≥ 0 and y² = x. If y = ∛x, then y³ = x. For negative x, the real cube root is negative; for example, ∛(−125) = −5.
Variables Explained
| Item | Meaning |
|---|---|
| Operation | Selects the principal square root or real cube root. |
| Radicand | The number placed under the radical sign and supplied to the selected operation. |
| Index | Two for a square root and three for a cube root. |
| Result | The real number that gives the radicand when raised to the selected index. |
Worked Example
Choose Cube Root and enter −125. Because (−5) × (−5) × (−5) = −125, the real cube root is −5. For comparison, choosing Square Root and entering 81 returns 9 because 9² = 81. Although −9 also squares to 81, the radical symbol √81 names the nonnegative principal root, 9.
Why Use This Calculator?
An integer answer indicates a perfect square or perfect cube for the chosen operation. Many inputs have irrational roots, so the displayed decimal is an approximation. For example, √2 continues beyond any fixed number of decimal places; the page shows 1.4142 after rounding. Keep more precision in scientific or financial work if later steps magnify rounding error.
A negative number has no real square root, which is why the square-root field rejects it. It does have complex square roots, but this page is limited to real results. Negative numbers do have real cube roots because multiplying three negative factors gives a negative product.
Roots also carry units. If an area is 144 square metres and represents a square, its side is √144 = 12 metres. If a cube has volume 125 cubic centimetres, its edge is ∛125 = 5 centimetres. Only take a root of a dimensional quantity when the underlying relationship supports it.
The operation choice matters more than the field label may suggest. For 64, the principal square root is 8 while the cube root is 4. For a decimal such as 0.008, the cube root is 0.2 because 0.2³ = 0.008. Scientific notation can help with very large or small values, but the browser field must accept the entry format. If a result is used in geometry, check whether the original quantity was squared or cubed and restore the corresponding linear unit. A root taken without that relationship may be numerically possible yet meaningless for the problem.
Common Mistakes
- Writing √25 as ±5; the radical symbol denotes the principal value 5, while solving x² = 25 gives two solutions.
- Trying to take a real square root of a negative number.
- Assuming ∛(−64) is positive; its real value is −4.
- Rounding a root early and reusing the shortened value in several later steps.
- Confusing a square root with dividing by two or a cube root with dividing by three.
Frequently Asked Questions
Why does √25 return 5 rather than −5?
The radical sign asks for the principal square root, defined as the nonnegative root. The equation x squared equals 25 has both 5 and −5 as solutions.
Can the calculator find a cube root of a negative number?
Yes. Odd roots preserve the sign, so the real cube root of −8 is −2.
Why is a decimal answer rounded?
The endpoint displays up to four decimal places. Many roots are irrational and cannot be written as a terminating decimal.
Is zero allowed?
Yes. Both √0 and ∛0 equal zero because zero raised to either power remains zero.
Can this page return complex roots?
No. It returns the principal real square root or the real cube root. Complex square roots require a different calculator.
How do I verify the answer?
Raise the result to the second power for a square root or the third power for a cube root. Allow for small differences caused by display rounding.
What is a perfect cube?
It is a number that equals an integer raised to the third power, such as 27 = 3³ or −64 = (−4)³.