Square Root Calculator
Compute square roots, cube roots, and nth roots with principal value, simplified radical form, and imaginary handling.
MATHCalculate the square root, cube root, or any nth root (up to n = 10) of a real number. Returns the principal root, the simplified radical form using the largest perfect-nth-power factor, and a decimal approximation. Even roots of positive numbers show both the positive and negative real solutions; even roots of negative numbers are handled as imaginary numbers using i = √-1.
The nth root of x is the value y such that y^n = x. For even n with positive x there are two real solutions (±y); the principal root is the non-negative one (this is what the radical symbol denotes). For odd n the result inherits the sign of x. Simplified radical form pulls the largest perfect-nth-power factor out from under the radical: for √72, the largest perfect square dividing 72 is 36, so √72 = √(36 × 2) = 6√2. Negative inputs to even roots produce imaginary numbers of the form bi. Worked example: √72 = 6√2 ≈ 8.485; ∛125 = 5; √-9 = 3i.
Square Root Calculator
Compute the square root, cube root, or any nth root of a number. Get the principal value, simplified radical form, and handle negative inputs with imaginary numbers when needed.
Roots, Radicals, and Imaginary Numbers
The nth root of x is the value y such that y^n = x. For even roots (square root, fourth root, etc.) of a positive number, there are actually two real solutions - one positive and one negative - because both (+y)^n and (-y)^n equal x. By convention the "principal root" is the non-negative one, which is what the radical symbol √ denotes. So √9 = 3 (principal), but the equation y^2 = 9 has two solutions: y = 3 or y = -3, often written as ±3. For odd roots there is only one real value, and it carries the sign of the input: ∛-8 = -2.
Simplified radical form rewrites a root by pulling out the largest perfect nth-power factor. For √72, factor 72 = 36 × 2 where 36 = 6^2 is the largest perfect square divisor. Then √72 = √(36 × 2) = √36 × √2 = 6√2. This is the exact value, while the decimal 8.4852... is only an approximation. Math teachers, the SAT, and proof-based courses generally want the simplified radical because it is exact and reveals structure. For nth roots, you look for the largest factor that is a perfect nth power: for ∛54, factor 54 = 27 × 2 = 3^3 × 2, so ∛54 = 3∛2.
Negative numbers under an even root have no real solution because any real number squared (or to any even power) is non-negative. Mathematicians extend the real numbers by defining the imaginary unit i = √-1, with the rule i^2 = -1. Then √-9 = √(9 × -1) = 3i, and √-72 = 6i√2. Numbers of the form a + bi are called complex numbers and form the foundation of electrical engineering (AC circuit analysis), quantum mechanics, signal processing (Fourier transforms), and large parts of pure mathematics. Negative inputs to odd roots stay real because an odd power of a negative number is still negative: ∛-27 = -3 with no imaginary part needed.
Educational math utility. Floating-point arithmetic introduces tiny rounding errors for very large numbers or high root indices - the simplified radical form is the exact value.