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A square root of a number x is a value that, when multiplied by itself, equals x. It is represented by the symbol √. For example: √9 = 3 because 3 × 3 = 9. To calculate a square root, you can use prime factorization (decompose into prime factors and group in pairs), successive approximation, or our online calculator which provides instant exact and decimal results with 8-decimal precision. Every positive number has two square roots: one positive (principal) and one negative, but by convention √ indicates the positive root.

A cube root of x is the value that, when multiplied by itself three times, equals x. Written as ∛x or x^(1/3). For example: ∛27 = 3 because 3 × 3 × 3 = 27. Unlike square roots, cube roots can be calculated for negative numbers in the real number system: ∛(-8) = -2 because (-2) × (-2) × (-2) = -8. This is because a negative number multiplied an odd number of times results in a negative number.

An nth root (ⁿ√x) is a value that, when raised to the power n, equals x. Can be written as x^(1/n). For example: ⁴√16 = 2 because 2⁴ = 16. Nth roots are used in various fields: calculating CAGR (compound annual growth rate) in finance with the formula (Final value/Initial value)^(1/n) - 1, normalizing vectors in game programming, calculating geometric means in statistics, and solving polynomial equations. Even-index roots require non-negative radicands, while odd-index roots work for any real number.

To simplify a radical, follow these steps: 1) Factor the radicand into prime factors using a factor tree, 2) Group factors by the root index (pairs for √, triplets for ∛), 3) Extract one factor from each complete group outside the radical, 4) Multiply extracted factors to get the coefficient, 5) Leave remaining factors under the radical. Example: √72 = √(2³ × 3²) = √(36 × 2) = 6√2. Always verify by squaring your answer: (6√2)² = 36 × 2 = 72 ✓

In the real number system, there is no square root of a negative number because no real number multiplied by itself gives a negative result (positive × positive = positive, negative × negative = positive). However, in the complex number system, √(-1) = i (imaginary unit). So √(-9) = √(9 × -1) = √9 × √(-1) = 3i. Complex numbers are used in electrical engineering, quantum physics, and signal processing. Cube roots and other odd-index roots can be calculated for negative numbers in the reals: ∛(-8) = -2.

Perfect squares are numbers that result from squaring an integer: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144, 169, 196, 225... (n² where n is an integer). Their square roots are always integers. Perfect cubes result from cubing an integer: 1, 8, 27, 64, 125, 216, 343, 512, 729, 1000... (n³). Recognizing these numbers helps with mental math and simplifying radicals. A number is a perfect square if all its prime factors appear an even number of times.

To multiply radicals with the same index, multiply the radicands: √a × √b = √(a×b). To divide, divide the radicands: √a ÷ √b = √(a/b). Examples: √2 × √8 = √16 = 4. √50 ÷ √2 = √25 = 5. √3 × √12 = √36 = 6. Always simplify the final result if possible. Note: You can only directly multiply or divide radicals that have the same index (all square roots, all cube roots, etc.).

Rationalizing means eliminating radicals from the denominator of a fraction. For simple fractions like 1/√2, multiply both numerator and denominator by √2: (1×√2)/(√2×√2) = √2/2. For expressions with binomials like 1/(√3+1), multiply by the conjugate (√3-1)/(√3-1) = (√3-1)/(3-1) = (√3-1)/2. This is standard practice in mathematics for expressing results in canonical form. Rationalized forms are easier to compare and use in further calculations.

Radicals and exponents are inverse operations, like multiplication and division. Raising to a power multiplies a number by itself multiple times (3² = 9), while taking a root undoes this operation (√9 = 3). They are related by the formula: ⁿ√x = x^(1/n). This means any root can be expressed as a power with a fractional exponent: √x = x^(1/2), ∛x = x^(1/3), ⁴√x = x^(1/4), etc. This relationship is fundamental in algebra and calculus.

Yes! Our root calculator is 100% free with no restrictions whatsoever. You can calculate square roots, cube roots, and nth roots without any limits. Get instant results with step-by-step solutions, automatic radical simplification, and 8-decimal precision. No registration, subscription, or installation required. Works on any device with a web browser: desktops, tablets, and smartphones. Perfect for students, teachers, engineers, and anyone who works with roots and radicals.