Pythagoras: right triangle with legs 5 and 12, hypotenuse?
Hypotenuse = √(5² + 12²) = √(25 + 144) = √169 = 13. Classic 5-12-13 Pythagorean triple. Square root is the most-used operation in geometry.
Root Calculator - Free Online Calculator
Calculate square root, cube root or any nth root of a number. Roots are the inverse of exponents - if x^n = y, then the nth root of y is x.
- Formula shown on the page
- Runs in your browser
- Your inputs stay private
2-th root of 2
1.41421356
Verification: result^2 ≈ 2
Understanding the Root Calculator
Roots are the inverse of powers - the nth root of a number x answers 'what value, when raised to the n-th power, gives x?'. The square root of 16 is 4 because 4² = 16; the cube root of 125 is 5 because 5³ = 125. This calculator computes square roots, cube roots, and any arbitrary nth root including fractional roots like 4√16 = 2 or 5√32 = 2. Used heavily in geometry (Pythagoras), physics (RMS values, periods of pendulums), engineering (rated values, square-root scaling), finance (volatility annualisation involves √12 for monthly returns), and statistics (standard deviation is the square root of variance).
Roots appear in nearly every Class 8-12 mathematics chapter - number system, algebra, geometry, mensuration, statistics. JEE/NEET aspirants encounter them in coordinate geometry distance formulas, trigonometric identities, projectile motion (range and time-of-flight involve square roots), and quadratic equation solutions. This calculator handles both the principal (positive) root and signals when roots aren't real-valued - even-indexed roots of negative numbers (like √(-9)) require complex numbers, which the calculator returns as NaN to flag the issue clearly.
Using the Root Calculator in 3 steps
Enter the number you want the root of.
Pick the root degree (2 = square root, 3 = cube root, n = custom).
Read the principal root.
How the Root Calculator works
Computed via x^(1/n). Even roots of negative numbers don't have real solutions and return NaN. Odd roots of negatives are well-defined and negative.
Formula
ⁿ√x = x^(1/n)
- ⁿ√xnth root of x
- xNumber under the radical (radicand)
- nRoot degree
How the Root Calculator gets its answer
Square root: √x = x^0.5 = the number whose square is x. Defined only for x ≥ 0 in real numbers. By convention, √x denotes the principal (non-negative) root, even though x = (±√x)² has two solutions ±√x. Example: √25 = 5 (principal), but x² = 25 has solutions x = ±5.
Cube root: ∛x = x^(1/3) = the number whose cube is x. Defined for all real x (positive, negative, zero) because the cube of a negative is negative. Example: ∛(-27) = -3 because (-3)³ = -27. This calculator handles negative radicands for odd-indexed roots correctly.
General nth root: ⁿ√x = x^(1/n). For even n (2, 4, 6...), requires x ≥ 0 for real solutions. For odd n (3, 5, 7...), works for any real x. Implementation: JavaScript computes x^(1/n) using Math.pow() which uses logarithms internally - precise to about 15 significant digits for typical inputs. Perfect roots (like ⁿ√(k^n) = k) return exact integer values; irrational roots return high-precision decimal approximations.
Worked examples
Quadratic formula: x² - 5x + 6 = 0
x = (5 ± √(25 - 24)) / 2 = (5 ± 1) / 2 = 3 or 2. Square root appears in every quadratic solution via the discriminant.
Volatility annualisation: monthly stock SD of 4%, annual?
Annual SD = monthly SD × √12 = 4% × 3.464 = 13.86%. Square root of time is the standard adjustment to scale variance across time periods in finance.
When to use the Root Calculator
- Class 8-12 mathematics - number system, algebra, geometry chapters.
- JEE Main, JEE Advanced, NEET - physics motion, geometry, coordinate problems.
- Pythagoras theorem - distance and hypotenuse calculations.
- Statistics - standard deviation (square root of variance).
- Finance - annualising volatility from monthly/weekly returns.
- Physics - RMS values, period of pendulums, fall times.
- Geometry - diagonal of rectangle, distance in 2D/3D space.
- Computer graphics - vector magnitude calculations.
Common mistakes and gotchas
- Taking even root of a negative number expecting real result. Even roots of negatives require complex numbers.
- Forgetting the ± when solving x² = y. Strictly, the equation has two solutions; √y is just the principal (positive) one.
- Confusing √(a+b) with √a + √b. They are NOT equal in general. √(9+16) = 5, but √9 + √16 = 3 + 4 = 7.
- Taking the root of a sum thinking it 'distributes'. Roots do not distribute over addition or subtraction.
- Reading ⁿ√x^m incorrectly. ⁿ√x^m = x^(m/n), useful for simplification.
Get more out of this tool
- Memorise perfect squares to 20 (1, 4, 9, 16, 25... 400). Speeds up mental √ calculation.
- Memorise perfect cubes to 10 (1, 8, 27, 64, 125, 216, 343, 512, 729, 1000). Speeds up cube root work.
- For estimation of √x where x is between two perfect squares, interpolate. √50 ≈ 7.07 (between 49 and 64).
- Use the relationship ⁿ√x = x^(1/n) when working with calculators or programming languages.
- Pair with the exponent, log and quadratic solver calculators on this site for related root-and-power math.
Questions about the Root Calculator
Why no real square root of −9?
Squaring any real number gives a non-negative result - there's no real x where x² = −9. Complex numbers introduce i = √−1 to handle this.
Cube root of −27?
−3, because (−3)³ = −27. Odd-degree roots of negatives are real.
What is the principal root?
The non-negative real root. Strictly, x² = 9 has two solutions ±3, but √9 by convention is the principal root +3.
How is the nth root calculated?
Most calculators use x^(1/n) which floating-point hardware computes via log + exp. For perfect squares/cubes the result is exact; for others it's a high-precision approximation.
What about fractional roots?
Like √(1/4) = 1/2. The calculator handles fractional radicands fine - just enter the decimal value.
Square root of zero?
0. Zero is the only number whose square is zero.
Math in your browser
From simple percentages to scientific notation, statistics, geometry and big-number arithmetic - every common math problem you'd reach for a calculator for, runs in your browser with no installs. The fraction calculator preserves exact rationals, the big-number calculator uses BigInt for hundreds of digits without precision loss, and the scientific calculator's expression evaluator is a hand-written shunting-yard parser (no eval) that supports trig, log, factorial and parentheses.
If you're a student, double-check your work without paying for a graphing calculator. If you're a professional, use these tools as a sanity check on spreadsheet formulas. Every result is shown alongside its formula and inputs, so the math is always auditable.
The Root Calculator is part of the Math suite on Online Calculator Tools, alongside 6 related calculators. All tools share the same UI patterns, the same privacy-first rendering (nothing is sent to a server), and the same commitment to showing the formula and assumptions behind every result.