√ Root Calculator
Compute square roots, cube roots, or any Nth root. Shows the decimal result, simplified radical form, and a verification. Handles negative numbers for odd-degree roots.
Root Formulas
Worked Examples
Example 1 — Simplify √72
Example 2 — Cube root of −27
Frequently Asked Questions
The square root of a number n is the value that when multiplied by itself equals n. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots: one positive and one negative.
A square root finds a value r such that r² = n, while a cube root finds r such that r³ = n. Cube roots can accept negative inputs (e.g. ∛(−8) = −2), whereas square roots of negative real numbers are undefined in the real number system.
Not in the real number system — the result would be an imaginary number (using the imaginary unit i where i² = −1). However, odd-degree roots (cube root, 5th root, etc.) of negative numbers are valid real numbers.
To simplify √n, find the largest perfect square that divides n. For example, √48 = √(16×3) = 4√3. Factor out the perfect square, take its root, and leave the remainder under the radical sign.
Related Calculators
Exponent Calculator
Calculate exponents, powers, and scientific notation instantly.
Logarithm Calculator
Calculate log base 10, natural log (ln), log base 2, and any custom base.
Pythagorean Theorem Calculator
Calculate the missing side of a right triangle using the Pythagorean theorem.