Root Calculator
Adjust the inputs below. Results update as you type.
How it works
√n = n^(1/2). ∛n = n^(1/3). ⁿ√x = x^(1/n). For non-perfect squares, results are irrational. Simplify √72 = √(36×2) = 6√2. Negative numbers have no real square root but have complex roots: √(−1) = i. Cube root of negative numbers is negative: ∛(−8) = −2. Newton's method converges quickly for numerical root approximation.
Input guidance
- Check units and decimal placement before running calculations.
- For percentage and ratio tasks, define the baseline value clearly.
- Use rounded output carefully and keep full precision for intermediate steps.
The formula
The nth root of x is the number that, raised to the power n, gives x: ⁿ√x = x^(1/n). The square root (n=2) and cube root (n=3) are most common.
Worked example
√144 = 12 because 12² = 144. The cube root of 27 is 3 because 3³ = 27.
More examples to test
- Baseline example: solve with clean numbers first to verify formula direction.
- Edge-case example: test near-zero or boundary values to avoid interpretation errors.
How to interpret results
Use outputs as quick checks, then verify with step-by-step work for graded or high-stakes use.
When this can be inaccurate
Errors usually come from unit mismatch, wrong baseline assumptions, or rounding too early in multistep problems.
Change history
- July 2026: Quality-reviewed for publication with formula checks and explanatory copy updates.
Site-wide corrections also appear on the corrections log.
Build math and logic skills with guided lessons
Educational platform recommendation.
Frequently asked questions
What is a square root?+
A value that multiplied by itself gives the original number. √49 = 7 because 7 × 7 = 49.
Can negative numbers have real roots?+
Even roots (square, fourth) of negatives are not real numbers, but odd roots are: the cube root of −8 is −2.
How does a fractional exponent relate to roots?+
x^(1/n) equals the nth root of x, so 8^(1/3) = 2. Roots and exponents are two views of the same operation.
Why do my classroom and calculator results differ?+
Differences usually come from rounding conventions, order-of-operations handling, or different baseline assumptions.
What is the safest way to validate an answer?+
Recompute with units shown and check against a second method or manual arithmetic for critical problems.