Square Root Calculator
Principal square root with exact-square detection, radical simplification (√72 = 6√2), and the verify-by-squaring check beside every answer.
Square Root Calculator
Results recalculate instantly on every keystroke. Nothing you type is transmitted.
What this result does not account for
- Real numbers only — no complex results
- No nested-radical simplification
In short: √144 = 12 exactly — a perfect square. √2 = 1.4142135623730951, irrational to the last digit. √72 simplifies to 6√2 because 72 = 36 × 2 and 36 is a perfect square. The symbol means the POSITIVE root: solving x² = 4 gives x = ±2, but √4 names only 2 — and √(−1) is refused, not imagined.
Formula
√x = y means y² = x, y ≥ 0
√(k² · m) = k√m
the radical symbol names the non-negative root — always.
Worked Example
- Detect the exact case. if the root is a whole number the page says so — 144 gives 12, not 12.000000.
- Simplify the radical. pull the largest square factor out: √72 = √(36 × 2) = 6√2.
- Verify. square the answer back — the check runs on every computation, not just the showcase ones.
Simplification is exact arithmetic, not rounding: 6√2 squares to 36 × 2 = 72 EXACTLY, while the decimal 8.485281… is only ever a portrait of it.
Strengths & Limits Of This Model
Where this engine is strong
- Exact-square detection — 144 prints as 12
- Radical form printed beside the decimal, never instead of it
Where it stops
- No continued-fraction expansion
- Single input — no expression parsing here
Practical Use Cases
Right triangles
the hypotenuse is a square root by the Pythagorean law
Standard deviations
variance becomes a distance by the square root
Area back-solves
a 72-square-foot square has 6√2-foot sides
Methodology & Editorial Standards
Compute the principal square root in double precision with Newton iteration; refuse negative inputs (no real square root). Perfect squares are detected by rounding and verified by squaring. For x > 1 the largest square factor k² is extracted so the radical prints as k√m; the decimal is trimmed to significant digits and the squaring check is printed beside it.
Computation runs in IEEE-754 double precision at full internal precision; rounding to two decimal places occurs strictly at the display layer, so no cumulative drift enters the result. All monetary outputs use accounting presentation — grouped thousands, two decimals, negatives in parentheses — so figures can be transcribed directly into a model or working paper. Division-by-zero and out-of-domain inputs return an em-dash rather than a misleading number.
This engine was reconciled against an independent reference implementation and hand-verified for the worked example above before release. Our full five-stage review process is published on the About Us page.
Disclaimer. This calculator is provided for informational and modelling purposes only and does not constitute financial, tax, legal, medical, or engineering advice. Verify all figures with a qualified professional before acting on them.
Square Root Calculator — 8 Expert FAQs
8 analyst-written answers to the questions practitioners actually ask — optimised for voice and answer-engine retrieval.
Why does √4 equal only 2, when (−2)² = 4 too?
The equation x² = 4 has two solutions, ±2 — but the radical SYMBOL is defined to name the non-negative one, the principal root. If both signs are wanted, write ±√4. One symbol, one value: that is what makes √x a function.
What does simplifying a radical actually do?
It rewrites the SAME number in factored clothing: √72 = √(36 × 2) = 6√2, because √(k·k × m) = k√m. Nothing is rounded — square 6√2 and you get exactly 72. The decimal 8.485281… is the portrait; the radical is the number.
Why is √2 irrational?
The classic proof: if √2 were a fraction a/b in lowest terms, then a² = 2b², so a² is even, so a is even — write a = 2k and b² = 2k² forces b even too. Both even contradicts lowest terms. No fraction fits; the decimal never repeats. This page prints the first sixteen digits of an endless, patternless tail.
What about the square root of a negative number?
Refused here, because no real number squares to a negative: squares are never below zero. The complex numbers answer with i, where i² = −1 — a different number system, and this page stays real. Odd roots are different: the Cube Root page accepts negatives by the sign law.
Is 0 a valid square root?
Yes: √0 = 0, the only value that is both a perfect square case and its own root. Zero is neither positive nor negative, and the principal-root rule includes it by definition (non-negative).
How does the page compute the root?
The engine iterates Newton’s method — guess, divide, average — which doubles correct digits every step, then verifies by squaring. Perfect squares are detected and printed as integers, so 144 gives 12, not 12.000000.
Square Root page or Root page?
The index-2 special case lives here with radical simplification; the general n-th root — fourth, fifth, and the odd-index sign law — lives on the Root page. They share the principal doctrine; they answer different indices.
Where do square roots show up unexpectedly?
Distances: the Pythagorean law makes every diagonal a square root. Volatility: standard deviation is the square root of variance. And the golden ratio is (1 + √5)/2 — an irrational born from a square root, like most numbers on the real line.