Nth Root Calculator
Finds the nth root of a number for any whole-number index — square root, cube root, or higher — by splitting off the largest perfect power that divides evenly, the same way a square root calculator simplifies √72 to 6√2.
How it works
How the Nth Root Calculator works
the part a result on its own leaves out
What an nth root undoes
The nth root of a number is whatever, raised to the nth power, gives that number back. The square root undoes squaring; the cube root undoes cubing; the pattern continues for any whole-number index. ∛8 is 2 because 2³ = 8.
Why some roots simplify and others don't
24's cube root simplifies to 2∛3 because 24 = 8 · 3, and 8 is a perfect cube. A number with no perfect-power factor above 1 — like 10's cube root — has nothing to split off, so it stays as an irrational radical rather than a whole number times a smaller radical.
Common mistakes
Common mistakes with nth roots
the errors that actually show up, not every possible slip
Taking an even root of a negative number
√−4 has no real value — no real number squares to a negative. An odd root of a negative number is fine: ∛−8 = −2, because (−2)³ = −8.
Splitting off a factor that isn't a perfect power for that index
∛24 does not simplify using the factor 4, because 4 is a perfect square, not a perfect cube. The factor pulled out has to be raised to the nth power itself — 8 = 2³ is what makes 2 come out of a cube root.
Mixing up the root index with the radicand
In ∛8, the small 3 is the index — which root — and 8 is the number the root is taken of. Swapping them, ⁸√3, asks a completely different question.
Questions
Asked often enough to answer here
answers that change how you work
What is the cube root of a negative number?
Real and negative. ∛−8 = −2, because −2 cubed is −8. This only works for odd roots — an even root of a negative number is not real.
How do I simplify a root that isn't a perfect power?
Find the largest factor that is a perfect power matching the root index, split it off, and leave the rest under the radical. ∛24 = ∛(8·3) = 2∛3, since 8 is a perfect cube.
What does the small number in front of the radical mean?
The root index — which root is being taken. No number shown means a square root by convention; a small 3 means a cube root, a small 4 a fourth root, and so on.
Is there a real 4th root of a negative number?
No. Any even root of a negative number is undefined over the real numbers, for the same reason a square root of a negative number is — no real number raised to an even power gives a negative result.
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