Root Calculator
Calculate square roots (√x), cube roots (∛x), and arbitrary nth roots (ⁿ√x) with exact radical simplification (kⁿ√r), prime factorization, and decimal values.
Root Calculator
Calculate square roots (√x), cube roots (∛x), and general nth roots (ⁿ√x) with exact radical simplification, prime factorization, and decimal approximations.
Square Root Calculator
Find principal and simplified square root of any real number: y = √x = x^(1/2)
Cube Root Calculator
Find exact simplified cube roots of positive and negative numbers: y = ∛x = x^(1/3)
General Root Calculator (nth Root)
Compute any arbitrary degree root: y = ⁿ√x = x^(1/n)
Understanding Roots and Radicals
In mathematics, the nth root of a number x is a value y which, when raised to the power of n, equals x:
- Radical Symbol (√): The mathematical symbol used to denote root extraction.
- Index / Degree (n): The superscript integer indicating which root is being computed. If no index is shown (e.g. √x), it implicitly represents the square root (n = 2).
- Radicand (x): The value positioned inside the radical symbol.
- Vinculum: The horizontal overbar extending across the top of the radicand.
Essential Properties & Rules of Radicals
The root of a product equals the product of the individual roots.
The root of a quotient equals the quotient of the numerator and denominator roots.
Roots convert directly to rational fractional exponents.
Nested radicals multiply their indices together.
Common Roots Reference Tables
Perfect Squares (1 to 20)
| Radicand (x) | Square Root (√x) | Radicand (x) | Square Root (√x) |
|---|---|---|---|
| 1 | 1 | 121 | 11 |
| 4 | 2 | 144 | 12 |
| 9 | 3 | 169 | 13 |
| 16 | 4 | 196 | 14 |
| 25 | 5 | 225 | 15 |
| 36 | 6 | 256 | 16 |
| 49 | 7 | 289 | 17 |
| 64 | 8 | 324 | 18 |
| 81 | 9 | 361 | 19 |
| 100 | 10 | 400 | 20 |
Perfect Cubes (1 to 10)
| Radicand (x) | Cube Root (∛x) | Negative (∛-x) |
|---|---|---|
| 1 | 1 | -1 |
| 8 | 2 | -2 |
| 27 | 3 | -3 |
| 64 | 4 | -4 |
| 125 | 5 | -5 |
| 216 | 6 | -6 |
| 343 | 7 | -7 |
| 512 | 8 | -8 |
| 729 | 9 | -9 |
| 1,000 | 10 | -10 |
How to Simplify Radicals Step-by-Step
To simplify a radical of the form n√x into its simplest radical form k n√r:
- Find the Prime Factorization of the Radicand: Break x into prime factors (e.g. 72 = 2³ × 3²).
- Group Factors into Powers of the Index (n): For square root (n = 2), group into pairs of identical factors: 72 = (2² × 3²) × 2 = (6²) × 2.
- Extract Perfect nth Powers Outside the Radical: Move the base of each complete power outside the radical sign: √72 = √(6² × 2) = 6√2.
- Multiply Remaining Factors Inside: Keep factors with exponents less than n inside the radical.
Frequently Asked Questions
What is the principal square root?
The principal square root is the non-negative square root of a non-negative real number. While the equation y² = 25 has two solutions (y = +5 and y = -5), the radical expression √25 uniquely denotes the principal positive value +5.
Can you take the square root of a negative number?
In the real number system (ℝ), no real number multiplied by itself gives a negative result. However, in the complex number system (ℂ), the square root of a negative number is an imaginary number using i = √(-1) (e.g. √(-25) = 5i).
Why can cube roots have negative real answers?
Because an odd number of negative factors produces a negative product: (-3) × (-3) × (-3) = -27. Therefore, the cube root of -27 is a real number equal to -3.
What happens if the root degree is zero (n = 0)?
The 0th root of a number (⁰√x = x^(1/0)) is mathematically undefined because it involves division by zero in the exponent.
Understanding Root Calculator & Formula
The Root Calculator on CalculatorScale.com is engineered to deliver instant, mathematically verified calculations conforming strictly to standard institutional equations. Whether planning long-term investments, managing household budgets, tracking clinical fitness targets, analyzing academic genetics, calculating veterinary dosages, or converting dimensional units, every calculation updates dynamically in real time without lag.
Every formula used in this tool adheres to certified international standards, including World Health Organization biometric metrics, veterinary clinical guidelines, NIST physical conversion constants, and standard biological equations. Results are computed entirely client-side inside your browser with complete data privacy, ensuring your sensitive numbers and statistics never leave your device.
Our calculators are 100% free with no registration, subscriptions, or paywalls required. You can toggle between imperial and metric units, customize input parameters, export calculation summaries, or embed this calculator directly into your own website or blog.
How to Use the Root Calculator
- 1 Select your calculation card: Square Root (√x), Cube Root (∛x), or General nth Root (ⁿ√x).
- 2 Enter the radicand number (x). For general root, also enter the root degree index (n).
- 3 Click the green "Calculate ▶️" button to evaluate.
- 4 Review the exact simplified radical form, prime factorization breakdown, decimal approximation, and both roots note.
Root Calculator — Frequently Asked Questions
Q1 What is the difference between principal square root and all roots? ▾
While the equation y² = 16 has two solutions (y = +4 and y = -4), the radical symbol √16 conventionally denotes only the principal positive square root (+4).
Q2 Can you take the root of a negative number? ▾
For odd indices (like cube root ∛(-27) = -3), the result is a real negative number. For even indices (like √(-25)), the result is an imaginary number (5i) in the complex number system.
Q3 How do you simplify radicals into simplest radical form? ▾
Factor the radicand into prime factors, group identical factors into powers of the index n, and pull the complete n-th power bases outside the radical sign.
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