Root Calculator
Take square roots, cube roots, or any nth root. For whole numbers it also simplifies the radical the way your math class wants it — √72 becomes 6√2 — and shows the perfect-power factoring behind it.
Example: with Root type Square root (√) · Number (x) 72 · Root degree (n) 4 → Root value: 8.485281374.
Computed by the calculator below using its default values. Change any input to see your own numbers.
Roots and radicals, briefly
The nth root undoes the nth power: because 6⁴ = 1,296, the 4th root of 1,296 is 6. Simplifying a radical means pulling every perfect-power factor outside the root sign: 72 contains the perfect square 36, so √72 = √(36 × 2) = 6√2. That exact form is preferred in algebra because the decimal (8.48528…) never terminates. Odd roots accept negative inputs (∛−8 = −2); even roots of negatives leave the real numbers entirely.
How it’s calculated
Decimal value = x^(1/n) (for negative x and odd n, computed as −|x|^(1/n)). Radical simplification, for integer x up to 10¹²: divide out each factor f while fⁿ divides x, collecting the outside coefficient — the largest k with kⁿ | x — leaving x = kⁿ·m and the answer k·ⁿ√m. Displayed to 10 significant digits.
Even roots of negative numbers are reported as complex (imaginary) rather than returning an error silently.
Worked example
√72: the largest perfect square inside 72 is 36, so 72 = 6² × 2 and √72 = 6√2 ≈ 8.485281. Likewise ∛54 = ∛(27 × 2) = 3∛2 ≈ 3.779763, and the 4th root of 1,296 is exactly 6.
Common mistakes
- Splitting roots over addition: √(9 + 16) = 5, not √9 + √16 = 7. Roots only split over multiplication.
- Simplifying with a factor that is not a perfect square (72 = 4 × 18 works, but 4 is not the largest — keep going to 36).
- Claiming √−72 = −8.49 — even roots of negatives are imaginary, not negative.
- Rounding the decimal too early and losing accuracy in later steps; keep the radical form until the end.
Frequently asked questions
How do I simplify a radical like √72?
Pull out the largest perfect-square factor: 72 = 36 × 2, and √36 = 6, so √72 = 6√2 ≈ 8.485. The calculator finds that factor automatically and shows the split; the same idea works for cube roots using perfect cubes (∛54 = 3∛2).
Can I take the root of a negative number?
Odd roots yes: ∛−8 = −2, because (−2)³ = −8. Even roots no — √−72 has no real answer (it is the imaginary number 6√2·i). The calculator returns odd roots of negatives and flags even ones as complex.
What is an nth root exactly?
The nth root of x is the number that gives x when raised to the nth power — equivalent to x^(1/n). So the 4th root of 1,296 is 6 because 6⁴ = 1,296. Fractional exponents and roots are the same operation written differently.
Why does the calculator show both a decimal and a radical form?
Most roots are irrational — their decimals never end — so 6√2 is exact while 8.485281 is an approximation. Math classes usually want the exact simplified radical; engineering and everyday use want the decimal. You get both.
Is the square root of a number always smaller than the number?
Only for numbers greater than 1. Between 0 and 1 the root is larger than the input: √0.25 = 0.5. And √1 = 1, √0 = 0 — the fixed points.