Root Calculator
Square, Cube & Nth Roots
Understanding Roots
A root answers the question "what number, raised to this power, gives me this result?" — the inverse operation of exponentiation.
The square root (n=2) and cube root (n=3) are the most common roots you'll encounter, from the Pythagorean theorem to volume calculations.
Odd roots (3, 5, 7…) always have exactly one real answer, even for negative inputs. Even roots (2, 4, 6…) of a negative number have no real answer at all.
Simplifying Radicals
\[ \sqrt{50} = \sqrt{25 \times 2} = \sqrt{25} \times \sqrt{2} = 5\sqrt{2} \] Breaking a number into a perfect-square factor and a remainder lets you express an irrational square root exactly, instead of rounding it to a decimal approximation.
Key Takeaways
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Odd roots of negative numbers are real.
∛(-8) = -2because(-2)³ = -8. - Even roots of negative numbers are not real. There is no real number that, squared, gives a negative result.
- Simplified radical form is exact, while a decimal result is always a rounded approximation of an irrational number.
Frequently Asked Questions
- Enter the number you want to find the root of.
- Enter the root index, or use the quick-select buttons for square root (2), cube root (3), or 4th root.
- Click "Calculate" to see the decimal result and, for square roots of positive integers, the simplified radical form.
Simplified radical form pulls the largest perfect-square factor out from under the root sign. For example, √50 = √(25 × 2) = 5√2, since 25 is a perfect square. This is the preferred way to write an irrational square root exactly, rather than as a rounded decimal.
A cube root (or any odd-indexed root) asks "what number, multiplied by itself 3 times, gives this result?" Since a negative number cubed stays negative ((-2)³ = -8), ∛(-8) = -2 is a valid real answer. A square root (or any even-indexed root) asks for a number that, multiplied by itself an even number of times, gives a negative — but any real number to an even power is non-negative, so no real answer exists.
It means the even root of a negative number has no solution among the real numbers. (Such an answer does exist among the complex numbers — √(-4) = 2i — but that's outside the scope of this calculator.)
The nth root of a number is the same as raising it to the power 1/n: ⁿ√b = b^(1/n). Our Exponent Calculator computes the reverse direction — powers instead of roots.