Square Root Calculator Online
Calculate square roots, cube roots, and nth roots with simplified radical form and decimal precision.
Input Parameters
Live Result
Details
Steps
Formula: āx = x^(1/2)
Calculate: 2ā(144) = 12
Quick Answer & Understanding Your Result
A Square Root Calculator determines the number that, when multiplied by itself, produces the original radicand: āx = r such that rĀ² = x. For negative values, the principal square root yields imaginary unit i (ā-x = iāx).
Common Principal Roots
- Perfect Squares: ā0 = 0, ā1 = 1, ā4 = 2, ā9 = 3, ā16 = 4, ā25 = 5, ā100 = 10, ā144 = 12.
- Irrational Roots: ā2 ā 1.41421, ā3 ā 1.73205, ā5 ā 2.23607, ā7 ā 2.64575.
- Radical Simplification: Factor out perfect square powers: ā72 = ā(36 Ć 2) = 6ā2.
Higher-Order & Complex Radicals
- Cube Roots (āx): Real roots exist for both positive and negative numbers (ā-27 = -3).
- Even Roots of Negatives: Produces imaginary and complex numbers (ā-16 = 4i).
- Inverse Power Notation: The nth root can be evaluated as fractional exponent x^(1/n).
Enter your radicand and degree into the calculator above to compute exact simplified radicals and decimal values.
Formula & Calculation Logic
Mathematical FormulaThe radical symbol ā denotes the principal square root. For any positive real number x, the principal square root is the unique non-negative real number r satisfying rĀ² = x.
Step-by-Step Guide
Select Root Degree
Choose root degree: 2 for square root, 3 for cube root, or enter custom n.
Enter Radicand
Input your number (positive, zero, or negative value).
Choose Decimal Precision
Select desired decimal places (from 2 up to 10 decimals).
View Radical & Decimal
Get the exact reduced radical form aāb, prime factorization, and decimal result.
Worked Examples Table
| Example | Input Values | Calculated Result | Explanation |
|---|---|---|---|
| Square Root of 144 | 12 Ć 12 = 144, so ā144 = 12. | ||
| Simplify Radical ā72 | Factor 72 into 36 Ć 2. ā36 = 6, yielding simplified form 6ā2. | ||
| Square Root of Negative ā-25 | ā-25 = ā25 Ć ā-1 = 5i. |
Frequently Asked Questions
Evidence-Based Methodology & Authoritative References
Wolfram MathWorld - Square Root
NIST Digital Library of Mathematical Functions
Standard: ISO / W3C / IETF Authoritative Domain Standard
Formula: āx = r such that rĀ² = x
Last Reviewed: August 21, 2026
Primary Source: Wolfram MathWorld - Square Root
Reference Document: NIST Digital Library of Mathematical Functions
Computation Privacy: 100% client-side 0ms computation, zero data retention
Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.
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Calculated results are modeled based on standard scientific and mathematical formulas. Please verify with domain-specific technical documentation for specialized applications.