Root Calculator
Calculate square roots, cube roots, and nth roots ⁿ√x.
√256 = 16
Roots are exponents in disguise
A root reverses a power: the square root of 49 is the number that squares to 49, and the cube root of 27 is 3. Written as exponents, √a = a^(1/2) and ∛a = a^(1/3), which is more than notation — it means every exponent rule applies to roots directly, with no separate system to learn.
Signs behave differently by index. Every positive number has two square roots, one positive and one negative, but the radical symbol denotes only the non-negative one — so√49 = 7, and −7 only appears when you write ±√49. Negative numbers have no real square root, since no real number squares to a negative. Odd roots have no such problem: ∛(−27) = −3 perfectly legitimately.
Simplifying means pulling out perfect-square factors: √72 = √(36 × 2) = 6√2. The rule works across multiplication and division — √(ab) = √a × √b — but not across addition. √(9 + 16) is 5, while √9 + √16 is 7. That mistake is common enough to be worth checking for deliberately.
Frequently Asked Questions
What is an nth root?
The nth root of a number x is a number r such that r^n = x. For example, 3rd root (cube root) of 27 is 3.
How do I calculate an nth root?
The nth root of x is x^(1/n). A cube root is x^(1/3), a fourth root x^(0.25). This is why roots obey the same rules as exponents, and why the root of a product equals the product of the roots.
Can you take the square root of a negative number?
Not within the real numbers, since any real squared is non-negative. Complex numbers define i as the square root of -1, making sqrt(-9) = 3i. Odd roots of negatives are real, however: the cube root of -8 is -2.
Why does a square root have two answers?
Because both 3 and -3 square to 9. By convention the radical symbol denotes only the non-negative principal root, which is why the quadratic formula writes the plus-or-minus explicitly - it would otherwise lose half the solutions.