Radical Calculator
Root index (n)
How to use this radical calculator
- Enter the radicand (the number under the root symbol) in the input field.
- Select the root index using the tabs: 2nd (square root), 3rd (cube root), 4th, or 5th.
- The result appears instantly with 10 decimal places of precision.
- For negative radicands, odd-index roots return real results while even-index roots are flagged as imaginary.
- Click Copy to grab the result for use in another calculation or spreadsheet.
Radical formula and key rules
Core formula
The radical formula is nth-root(x) = x^(1/n). Every radical can be expressed as a fractional exponent. This is how the calculator computes results: by applying the 1/n exponent to the radicand. For x = 625 and n = 4: 625^(1/4) = 5, because 5^4 = 625.
Product rule
nth-root(a x b) = nth-root(a) x nth-root(b) -- radicals distribute over multiplication. This rule is essential for simplification. Example: square root of 72 = square root of (36 x 2) = 6 x square root of 2. Always factor the radicand and pull out the largest perfect-power factor first to reach simplest form in one step.
Quotient rule
nth-root(a / b) = nth-root(a) / nth-root(b) -- radicals distribute over division too. Example: square root of (25/16) = 5/4 = 1.25. Useful when numerator and denominator are both recognizable perfect powers -- produces an exact fraction without any decimal conversion.
Negative radicands
For odd indices (3, 5, 7...), negative radicands produce real results: cube root of -27 = -3. For even indices (2, 4, 6...), negative radicands produce imaginary results. The key rule: (-1)^(1/n) is real only when n is odd. This calculator correctly flags the imaginary case and computes real results for all other inputs.
Quick reference
| Rule | Formula | Example |
|---|---|---|
| Core formula | nth-root(x) = x^(1/n) | 4th-root(81) = 3 |
| Product rule | nth-root(a*b) = nth-root(a)*nth-root(b) | sqrt(72) = 6*sqrt(2) |
| Quotient rule | nth-root(a/b) = nth-root(a)/nth-root(b) | sqrt(25/16) = 5/4 |
| Odd negative | nth-root(-x) = -nth-root(x) (odd n) | cbrt(-27) = -3 |
| Even negative | nth-root(-x) = imaginary (even n) | sqrt(-4) undefined in R |
What is a radical in mathematics?
A radical is any expression written using the root symbol, also called a radical sign. The number under the root symbol is the radicand, and the small superscript number to the upper left of the radical is the index. When no index is shown, it defaults to 2 (square root). Radical expressions are fundamental in algebra, geometry, and every field that involves solving for unknown sides, frequencies, or growth rates.
Every radical can be converted to a fractional exponent: the nth root of x equals x raised to the power 1/n. This equivalence is critical in calculus because power rules apply directly to fractional exponents, making it possible to differentiate and integrate expressions like the square root of x without special radical-specific techniques. The calculator above applies the 1/n exponent precisely for every supported index.
Radicals appear constantly in applied math. The square root appears in the Pythagorean theorem (finding triangle sides), distance formulas, standard deviation in statistics, and the RMS voltage in electronics. Cube roots appear in volume-to-side conversion and finance (CAGR). The 4th and higher roots appear in material science, acoustics, and information theory. Mastering the product and quotient rules for radicals lets you simplify these expressions before reaching for a calculator.
- sqrt(25) = 5 because 5 x 5 = 25 (perfect square)
- cbrt(64) = 4 because 4 x 4 x 4 = 64 (perfect cube)
- 4th-root(256) = 4 because 4^4 = 256
- cbrt(-125) = -5 because (-5)^3 = -125 (odd index, real result)
Interpreting your radical result
Perfect power -- exact integer result
When flagged as a perfect power, the result is an exact integer with no rounding. The square root of 144 = 12 exactly, the cube root of 512 = 8 exactly. Perfect-power results come up constantly in geometry, algebra, and exam problems where exact answers are required.
Non-perfect -- irrational decimal
Most radicands are not perfect powers. Their roots are irrational -- the decimal expansion never ends or repeats. The calculator shows 10 decimal places. The step-by-step section verifies accuracy by raising the rounded result to the power n and comparing to the original radicand.
Imaginary result (even index, negative radicand)
Square roots, 4th roots, and all even-index roots of negative numbers are imaginary. They involve the unit i (where i^2 = -1) and cannot be shown on the real number line. The calculator flags these inputs. For real results with negative radicands, switch to an odd index (3, 5, 7...) which always produces real, negative results.
Real-world applications of radical expressions
Geometry -- Pythagorean theorem
The Pythagorean theorem states a^2 + b^2 = c^2 for a right triangle, so the hypotenuse c = square root of (a^2 + b^2). A triangle with legs 3 and 4 has hypotenuse = square root of (9 + 16) = square root of 25 = 5. Construction workers, architects, and surveyors use this formula daily to verify right angles and calculate diagonal distances.
Statistics -- standard deviation
The standard deviation of a dataset is the square root of the variance. If a sample has variance 81, the standard deviation = square root of 81 = 9. The square root converts squared units back to original units: if you measure heights in cm, variance is in cm^2 but standard deviation comes back in cm. This is the most common radical in data analysis, finance, and quality control.
Physics -- pendulum period
The period of a simple pendulum is T = 2pi x square root of (L/g), where L is the length and g is gravitational acceleration. A pendulum 1 meter long has period T = 2pi x square root of (1/9.81) approximately 2.006 seconds. Physicists, clockmakers, and seismic instrument designers all use this radical formula to predict oscillation.
Electrical engineering -- RMS voltage
The root mean square (RMS) voltage of a sinusoidal signal is peak voltage divided by square root of 2. A household outlet rated at 120V RMS has a peak voltage of 120 x square root of 2 approximately 169.7V. The square root of 2 appears in every AC circuit analysis, power calculation, and signal amplitude conversion in electrical engineering.
Common radical values reference table
| Radicand | Square root | Cube root | 4th root |
|---|---|---|---|
| 1 | 1.000 | 1.000 | 1.000 |
| 2 | 1.414 | 1.260 | 1.189 |
| 4 | 2.000 | 1.587 | 1.414 |
| 8 | 2.828 | 2.000 | 1.682 |
| 9 | 3.000 | 2.080 | 1.732 |
| 16 | 4.000 | 2.520 | 2.000 |
| 25 | 5.000 | 2.924 | 2.236 |
| 27 | 5.196 | 3.000 | 2.280 |
| 36 | 6.000 | 3.302 | 2.449 |
| 64 | 8.000 | 4.000 | 2.828 |
| 100 | 10.000 | 4.642 | 3.162 |
| 125 | 11.180 | 5.000 | 3.344 |
Quick radical calculation tips
- Perfect squares to memorise: 1, 4, 9, 16, 25, 36, 49, 64, 81, 100, 121, 144. Instantly recognise when an exact integer root is possible.
- Factor first: Before computing, factor the radicand for perfect-power factors. sqrt(300) = sqrt(100 x 3) = 10 x sqrt(3). Easier to work with.
- 4th root = sqrt twice: 4th-root(x) = sqrt(sqrt(x)). Works on any calculator with a square root button.
- Fractional exponent key: Use x^(1/n) on a scientific calculator to compute any root without a dedicated radical button.
- Negative check: If your radicand is negative, verify your root index is odd (3, 5, 7...) before expecting a real result.
Common radical mistakes to avoid
Assuming sqrt(-x) is always imaginary
Square roots of negative numbers ARE imaginary. But cube roots and other odd-index roots of negatives are real. Students who learn "you cannot take the root of a negative number" sometimes over-apply this rule. The correct statement is: "even-index roots of negative numbers are imaginary." Cube root of -64 = -4 is perfectly real and correct.
Misapplying the product rule across addition
The product rule applies to multiplication under the radical, NOT addition. sqrt(a + b) is NOT equal to sqrt(a) + sqrt(b). Example: sqrt(9 + 16) = sqrt(25) = 5, NOT sqrt(9) + sqrt(16) = 3 + 4 = 7. This is one of the most common algebra errors. The radical of a sum cannot be split -- only the radical of a product can be factored.
Not simplifying before computing
For exact answers in algebra, always simplify the radical before converting to a decimal. sqrt(72) simplified is 6 x sqrt(2), not 8.485. Using 8.485 in further calculations accumulates rounding error. Simplified exact form is always preferred in symbolic math, proofs, and any situation where you need to keep expressions exact.
Dropping the negative sign on odd-index roots
The cube root of -125 is -5, not +5. Many calculators only show the positive (principal) root for even indices, but for odd indices the sign of the radicand carries through to the result. Always check the sign: nth-root(-x) = -nth-root(x) for odd n. The step-by-step section above shows the sign explicitly for all negative-radicand computations.