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Free online root calculator to calculate square roots, cube roots, nth roots, and radical expressions. Get instant results with step-by-step solutions for all your root calculations.
2 = Square Root, 3 = Cube Root, 4 = Fourth Root, etc.
A root (also called a radical) is the inverse operation of exponentiation. The nth root of a number x is a value that, when raised to the power n, equals x. Mathematically, if y = โฟโx, then yโฟ = x.
The most common types of roots are:
The symbol โ is called the radical sign, the number under it is the radicand, and the small number indicating which root is the index. When no index is shown, it's assumed to be 2 (square root).
Input the number (radicand) for which you want to find the root. Can be positive, negative (for odd roots), or decimal.
Enter the root index: 2 for square root, 3 for cube root, 4 for fourth root, or any positive integer.
Click the calculate button to get instant results with high precision decimal values.
Review your result with step-by-step explanations and verification of the calculation.
The most common root. A number that when multiplied by itself gives the original number.
Examples: โ16 = 4 (because 4ร4 = 16), โ25 = 5, โ100 = 10
A number that when multiplied by itself three times gives the original number. Can be negative.
Examples: ยณโ8 = 2 (because 2ร2ร2 = 8), ยณโ27 = 3, ยณโ(-8) = -2
A number that when raised to the 4th power gives the original number.
Examples: โดโ16 = 2 (because 2โด = 16), โดโ81 = 3, โดโ256 = 4
General form for any root. If y = โฟโx, then yโฟ = x.
Examples: โตโ32 = 2 (because 2โต = 32), โถโ64 = 2, ยนโฐโ1024 = 2
โฟโ(a ร b) = โฟโa ร โฟโb
Example: โ(4ร9) = โ4 ร โ9 = 2ร3 = 6
โฟโ(a / b) = โฟโa / โฟโb
Example: โ(16/4) = โ16 / โ4 = 4/2 = 2
โฟโ(aแต) = (โฟโa)แต = a^(m/n)
Example: โ(xยณ) = x^(3/2)
โฟโa = a^(1/n)
Example: โx = x^(1/2), ยณโx = x^(1/3)
แตโ(โฟโa) = แตโฟโa
Example: ยณโ(โ64) = โถโ64 = 2
Odd roots can be negative
Example: ยณโ(-8) = -2, but โ(-4) is complex
Finding side lengths from area or volume. Pythagorean theorem: c = โ(aยฒ+bยฒ)
Compound interest calculations, finding interest rates: r = โฟโ(FV/PV) - 1
Velocity calculations, pendulum periods, free fall: v = โ(2gh)
RMS values, stress analysis, structural calculations
Standard deviation, geometric mean, variance calculations
Distance formula, vector normalization, 3D transformations
Calculate powers and exponents
Advanced mathematical operations
Logarithm calculations
Solve quadratic equations
Add, subtract, multiply fractions
Calculate percentages easily
Simplify and calculate ratios
Find factors of numbers
Comprehensive lessons on square roots, cube roots, and radical expressions
Visit Khan Academy โEasy-to-understand explanations with examples and practice problems
Visit Math is Fun โCalculate any root index - not just square and cube roots. Support for 5th, 7th, 10th, or any root you need.
Detailed explanations of how the root is calculated, helping you learn and verify your work.
Results with up to 10 decimal places for accurate scientific and engineering calculations.
Calculate odd roots of negative numbers (like ยณโ-8 = -2) with proper handling of complex results.
Automatic simplification to radical form using prime factorization for cleaner mathematical expressions.
Calculate roots, simplify radicals, convert decimals, and rationalize denominators - all in one tool.
Track your recent calculations with timestamps for easy reference and comparison.
Works seamlessly on all devices - desktop, tablet, and mobile with optimized touch interface.
This root calculator was developed by CalcFinex, a team dedicated to creating accurate, educational mathematical tools. Our calculator implements industry-standard algorithms for root calculation, including the Newton-Raphson method for high-precision results and prime factorization for radical simplification.
Mathematical Accuracy: All calculations are performed using JavaScript's built-in Math.pow() function with IEEE 754 double-precision floating-point arithmetic, providing accuracy up to 15-17 significant decimal digits. Results are verified against the inverse operation (exponentiation) to ensure correctness.
Educational Purpose: This tool is designed for students, educators, engineers, and professionals who need reliable root calculations. The step-by-step solutions help users understand the mathematical process, not just get answers. All formulas and methods align with standard mathematical textbooks and curriculum guidelines.
Regular Updates: We continuously improve this calculator based on user feedback and mathematical best practices. The tool is tested against known mathematical identities and edge cases to ensure reliability across all input ranges.
Note: While this calculator provides highly accurate results, for critical applications in engineering, finance, or scientific research, always verify results with multiple sources and consider the limitations of floating-point arithmetic for extremely large or small numbers.