√ Root Calculator

Find square roots, cube roots, and nth roots of positive numbers with exact and decimal results.

Square, Cube, and nth Root Extraction

BrainyCalculators editorial insight — unique to this tool

√144 = 12; ∛27 = 3. Geometry uses √ for diagonals (square side s → diagonal s√2). Engineers use nth roots in damping ratios and filter design. Negative numbers have complex even roots — calculator may show real-only results.

When to use this calculator

Use for root evaluation. For general powers, use Exponent.

Computing powers rather than roots?

This page extracts nth roots. For base-to-power evaluation, use the Exponent Calculator →

Number (n)

Root Degree

What is a Root Calculator?

A root calculator computes √x, ∛x, and general nth roots. It answers “what number raised to n equals x?” for positive radicands.

Use this page for geometry side lengths, RMS values, and manual root checks. It evaluates roots numerically rather than simplifying radical expressions symbolically.

For b^n power problems, use the Exponent Calculator. For equation solving with variables, use the Algebra Calculator.

Root Formulas

Nth root: ⁿ√x = x^(1/n)

Simplify √n: find largest perfect square factor k², then √n = k√(n/k²)

Negative (odd degree): ⁿ√(−x) = −ⁿ√x when n is odd

Worked Examples

Example 1 — Simplify √72

72 = 36 × 2 → √72 = √36 × √2 = 6√2 ≈ 8.485

Example 2 — Cube root of −27

∛(−27) = −∛27 = −3

How the Root Calculator Works

Formula, assumptions, and calculation steps for this math tool.

Methodology

Math calculators apply the relevant arithmetic, algebraic, geometric, or numeric rule to the values entered and simplify the result where possible.

Calculation Steps

Read the values and operation selected.
Normalize signs, decimals, fractions, or units if needed.
Apply the mathematical rule or formula.
Format the answer and any intermediate values for checking.

Assumptions and Limits

Inputs must be within the supported domain of the operation.
Decimal answers may be rounded for readability.
Symbolic simplification is limited to the calculator scope.

Frequently Asked Questions

What is a square root?

The square root of a number n is the value that when multiplied by itself equals n. For example, √25 = 5 because 5 × 5 = 25. Every positive number has two square roots: one positive and one negative.

What is the difference between square root and cube root?

A square root finds a value r such that r² = n, while a cube root finds r such that r³ = n. Cube roots can accept negative inputs (e.g. ∛(−8) = −2), whereas square roots of negative real numbers are undefined in the real number system.

Can you take the square root of a negative number?

Not in the real number system — the result would be an imaginary number (using the imaginary unit i where i² = −1). However, odd-degree roots (cube root, 5th root, etc.) of negative numbers are valid real numbers.

How do you simplify a radical?

To simplify √n, find the largest perfect square that divides n. For example, √48 = √(16×3) = 4√3. Factor out the perfect square, take its root, and leave the remainder under the radical sign.

Real-World Applications

📈

Compound Annual Growth Rate (CAGR)

CAGR = (End Value / Start Value)^(1/n) − 1 uses the nth root to convert a multi-year total return into an equivalent annual growth rate. A $10,000 investment that grew to $16,105 over 5 years has a CAGR of (16105/10000)^(1/5) − 1 = 10% per year — the key metric for comparing investment performance across different time horizons.

📐

Hypotenuse & Distance Calculation

The square root in the Pythagorean theorem (c = √(a² + b²)) and the 2D/3D distance formula (d = √(Δx² + Δy² + Δz²)) is the most applied root calculation in geometry, physics, and computer programming — used in GPS distance calculation, game collision detection, and engineering structural analysis.

📊

Standard Deviation in Statistics

Standard deviation = √(variance) — the square root converts variance (which is in squared units) back into the original unit of measurement, making it interpretable. A dataset with variance of 25 kg² has a standard deviation of 5 kg, directly comparable to the mean in the same units. Standard deviation is foundational to statistical inference, quality control, and risk management.

📦

Cube Volume to Side Length

Given a cube with volume V, the side length is the cube root: side = ∛V. A storage box with volume 8,000 cm³ has side length ∛8000 = 20 cm. This calculation appears in packaging design, warehouse optimisation, and material science when converting volumetric measurements to linear dimensions.

⚡

RMS Voltage & Signal Processing

Root Mean Square (RMS) voltage is the square root of the mean of squared voltage values over a cycle — the effective value of an AC voltage that would deliver the same power as a DC voltage. UK mains voltage is 230V RMS from a 325V peak. RMS calculations are fundamental in electrical engineering, audio signal processing, and vibration analysis.

🔬

Radioactive Decay Half-Life Calculations

The fraction remaining after n half-lives is (1/2)^n — finding the number of half-lives elapsed from a remaining fraction requires the logarithm, while finding the fraction after a fractional half-life uses fractional exponents (nth roots). These calculations underpin radiocarbon dating, nuclear medicine dosing, and nuclear waste storage planning.

Common Mistakes

Taking the square root of a negative number without expecting a complex result

The square root of a negative number is not a real number — it is imaginary. √(−9) = 3i, where i = √(−1). Attempting to compute the square root of a negative value in real number arithmetic (e.g., in a spreadsheet without complex number handling) produces an error or #NUM! result. In physics and engineering, negative values under a square root indicate an impossibility in the real domain — such as imaginary frequency or no real solution to the quadratic formula.

Confusing the square root with squaring

A common calculator error is squaring a number (x²) when the square root (√x) was intended, or vice versa. These are inverse operations: √(9) = 3, while 9² = 81. When computing the hypotenuse of a right triangle, the correct operation is to take the square root of the sum of squares — not to square the sum. Verify that the calculator operation matches the intended mathematical operation.

Forgetting that even-index roots of negative numbers are undefined in real arithmetic

Square roots (n=2) and all even-index roots (n=4, 6, 8...) of negative numbers are undefined in the real number system. Odd-index roots (cube root n=3, fifth root n=5) of negative numbers are real and negative: ∛(−27) = −3. In real-world applications, a negative value appearing under an even-index root typically signals a data error, incorrect formula setup, or an unphysical scenario.

Rounding intermediate root calculations before the final step

When a root calculation feeds into a subsequent calculation, rounding the root result prematurely introduces error that compounds through subsequent operations. For example, computing √2 ≈ 1.41 (2 decimal places) instead of 1.41421356... then multiplying by 100 gives 141 instead of 141.42 — an error of 0.3%. In chained calculations, maintain full precision throughout and only round the final result.

Applying the square root to both sides of an inequality incorrectly

While √(x²) = |x| (the absolute value of x), students frequently write √(x²) = x, losing the negative case. For instance, solving x² = 9 gives x = ±3 (not just x = 3) because both +3 and −3 squared equal 9. In geometry, length is always positive so only the positive root is taken — but in algebra, both roots must be considered unless physical constraints eliminate the negative solution.

Common Roots Quick Reference

Root Type	Notation	Example
Square root (n=2)	√x or x^(1/2)	√144 = 12
Cube root (n=3)	∛x or x^(1/3)	∛125 = 5
Fourth root (n=4)	⁴√x or x^(1/4)	⁴√81 = 3
nth root (general)	ⁿ√x or x^(1/n)	⁵√32 = 2
CAGR (investment)	(End/Start)^(1/n) − 1	(1.6105)^(1/5) − 1 = 10%

References

Stewart, J. Calculus: Early Transcendentals. Cengage, 2016.
Axler, S. Algebra and Trigonometry. Wiley, 2021.
Abramowitz, M. and Stegun, I.A. Handbook of Mathematical Functions. National Bureau of Standards, 1964.
NIST. Digital Library of Mathematical Functions. dlmf.nist.gov, 2024.
Knuth, D.E. The Art of Computer Programming, Vol. 1. Addison-Wesley, 1997.