Square Root Calculator
Roots computed with IEEE-754 double precision (Math.sqrt, Math.cbrt). Radical form uses prime factorization — e.g. √72 = 6√2. Related: exponent, factor, Pythagorean theorem.
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What Your Result Means
- Square Root (√): The number that, when multiplied by itself, gives your input. For example, √25 = 5 because 5 × 5 = 25.
- Cube Root (∛): The number that, multiplied by itself three times, gives your input. Useful for volume-related problems.
- Perfect Square: If "Yes," your number is the exact square of an integer — its square root is a whole number with no remainder.
- Simplified Radical: Expresses the square root in simplest form by pulling perfect-square factors outside the radical sign (e.g., √72 = 6√2).
- Prime Factorization: Your number broken into its prime building blocks, which is the basis for simplifying radicals.
How This Calculator Works
You enter any non-negative number. The tool computes square, cube, and fourth roots using JavaScript's built-in Math functions (IEEE-754 double precision). For the simplified radical, it runs a prime factorization, pairs up factors, pulls pairs outside the radical sign, and leaves the remainder under the root. Results are rounded to six decimal places.
Quick Questions
Can I find the square root of a negative number?
Not with real numbers — the square root of a negative number is imaginary. This calculator handles non-negative inputs only. For complex roots, the result would involve the imaginary unit i (e.g., √−4 = 2i).
What does "simplified radical form" mean?
It means rewriting a square root so no perfect-square factor remains under the radical. For example, √72 = √(36 × 2) = 6√2. This is the standard form expected in algebra and geometry classes.
Why does the calculator show 6 decimal places?
Six decimal places give enough precision for most practical and homework purposes while avoiding misleading extra digits. JavaScript's floating-point math is accurate to about 15–16 significant digits internally.
How is prime factorization related to square roots?
To simplify √n, you factor n into primes, group them in pairs, and move each pair outside the radical as a single factor. Unpaired primes stay inside. That's why 72 = 2³ × 3² simplifies to 6√2.
Sources
- Wikipedia — Square Root (definition, properties, computation methods)
- Khan Academy — Simplifying Square Roots (step-by-step radical simplification)
Method & review
Estimate only. Results reflect your inputs and standard formulas. Double-check important decisions independently.