Root Calculator
Calculate square roots, cube roots, and nth roots of numbers. Free, fast, accurate — no signup, mobile-friendly, works i
Root Type
Parameters
The number to find the root of
The index of the root (2 for square root, 3 for cube root, etc.)
📐 Common Roots
Root Rules
Primary Result
Root Value
Perfect root
Type
Rational
Scientific
notation
Squared
value²
Cubed
value³
Verification: Correct
Step-by-Step Solution
How it works
A root calculator finds the nth root of a number — the value that, raised to the nth power, gives back the number. The square root (n=2) and cube root (n=3) are the most common; an nth root is the same as raising to the power 1/n.
nth root
ⁿ√x = x^(1/n) such that (ⁿ√x)ⁿ = x
- x
- the number (radicand)
- n
- the root index (2 = square, 3 = cube)
Worked example
- Find the cube root of 64
- Which y gives y³ = 64?
- 4 × 4 × 4 = 64
³√64 = 4.
Good to know
- Odd roots of negative numbers are real (³√−8 = −2); even roots of negatives are not.
- A fractional exponent and a root are the same thing: x^(1/3) = ³√x.
- Most roots are irrational and can only be approximated as decimals.
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Frequently Asked Questions
What is an nth root?
The nth root of x is the number that, multiplied by itself n times, gives x — written as x^(1/n). The 4th root of 81 is 3 because 3 x 3 x 3 x 3 = 81.
Can I take the root of a negative number?
Odd roots of negatives are real: the cube root of -27 is -3. Even roots of negative numbers (square, 4th, etc.) have no real solution and require imaginary numbers.
How do roots relate to fractional exponents?
A root is just a fractional exponent: x^(m/n) equals the nth root of x raised to the m. For example, 8^(2/3) is the cube root of 8 (which is 2) squared, giving 4.
How do I simplify a radical?
Factor out perfect powers matching the root index. For square roots, √72 = √(36 x 2) = 6√2; for cube roots, ∛54 = ∛(27 x 2) = 3∛2.
How can I estimate a root without a calculator?
Bracket the answer between perfect powers and interpolate: ∛50 lies between ∛27 = 3 and ∛64 = 4, closer to 3.7. Repeated averaging (Newton's method) refines the estimate quickly.