Simplify √n into a√b form — with the full factoring steps shown.
√72 = 6√2 — the messy root tamed into a clean coefficient times a small radical. Simplifying radicals factors out perfect squares from under the root, leaving the simplest exact form: the form textbooks demand, exact where decimals approximate. Our free Simplifying Radicals Calculator factors the radicand, pulls out square factors step by step, and shows the decimal check.
The rule is √(ab) = √a × √b: split the radicand into a perfect square times a remainder, root the square, and leave the rest. √72 = √(36×2) = √36 × √2 = 6√2. Every “simplify” problem is a hunt for the largest square factor.
Enter any non-negative integer and the calculator returns the a√b form with the factoring chain and decimal verification. Whether you are doing algebra homework, rationalizing denominators, or checking exact forms, the steps are shown and the math is exact.
A radical is in simplest form when the radicand has no perfect-square factors (other than 1): 6√2 is simple; √72 is not, because 72 = 36 × 2 and 36 is a square. Simplification extracts those squares: √72 = √(36×2) = 6√2.
Why exact form matters: 6√2 is exact; 8.4853 is rounded. In multi-step algebra the rounding error compounds, while the exact form stays clean — and teachers grade the form, not just the value. The widget shows both: exact form big, decimal as the check.
The algorithm: factor the radicand into primes; pair up identical primes (each pair is a square that comes out); unpaired primes stay under the root. 72 = 2³ × 3² → one pair of 2s (a 2 comes out, one 2 stays), one pair of 3s (a 3 comes out) → 2×3√2 = 6√2.
Key terms:
Only like radicals combine by addition — 3√2 + 5√2 = 8√2, but √2 + √3 cannot combine. Simplification’s payoff is revealing like radicals.
Put √72 in simplest radical form.
The canonical example: every textbook simplifies √72 first, because 36 is the giveaway square.
Put √200 in simplest radical form.
Same leftover √2 as √72 — so √72 + √200 = 16√2. Simplification reveals combinability.
Sofia must compute √48 + √27. Unsimplified, they cannot combine. She simplifies: √48 = √(16×3) = 4√3; √27 = √(9×3) = 3√3. Like radicals! Sum: 7√3. The widget’s step chains did both factorizations, and the homework’s trick — hidden like radicals — is exposed.
A second scenario: rationalizing 1/√18. Simplify first: √18 = 3√2, so 1/(3√2) = √2/6 after rationalizing. The simplification halves the work.
A third: a geometry problem gives a hypotenuse of √98. √98 = √(49×2) = 7√2 — the exact form the answer key expects, not 9.8995.
A fourth: checking √50 × √8. Multiply first: √400 = 20 — or simplify: 5√2 × 2√2 = 10 × 2 = 20. Both routes, same exact answer.
Algebra: simplest radical form is the standard exact answer — quadratic formula outputs, distance formula results, and completing-the-square all end in radicals that must be simplified before they’re “done.”
Geometry: diagonals (√2 per unit), space diagonals (√3), and 30-60-90 triangles (1 : √3 : 2) express exact lengths as simplified radicals — decimals would forfeit exactness.
Trigonometry: exact trig values are radicals — sin 75° = (√6 + √2)/4. Simplification keeps these recognizable; unsimplified forms hide the special angles.
Calculus: derivatives of roots and integrals producing radicals all simplify the same way — and limits at infinity often hinge on rationalizing, the reciprocal skill.
Number theory: squarefree kernels, quadratic residues, and continued fractions all factor radicands the way the widget does — prime factorization is the universal key.
Physics: RMS values, pendulum periods, and wave speeds carry roots that combine only when simplified to like radicals — 3√2 + 5√2 = 8√2 is a physics calculation, not just algebra.
Computer algebra: systems like SymPy canonicalize radicals by exactly this algorithm — extract square factors, sort, combine likes. The widget is a window into CAS internals.
One pattern worth internalizing: simplify before combining. Addition needs like radicals, and only simplification reveals them. It is the same instinct as finding common denominators before adding fractions.
Electrical engineering: impedance calculations produce √2 factors (RMS from peak: Vrms = Vpeak/√2). Simplified radicals keep the phasor math exact through multi-stage filter designs.
Computer graphics: normalizing the vector (3, 3) divides by √18 = 3√2 — simplified first, the division stays clean. Game engines simplify symbolically for the same reason students do.
Architecture: the √2 rectangle proportions of A-series paper (1:√2) come from the unit square’s diagonal — a simplified radical as an ISO standard (ISO 216).
Contest math: √2 + √8 + √18 = √2 + 2√2 + 3√2 = 6√2 — the entire genre of “combine the radicals” problems is simplification-then-addition. The widget automates the first half.AC circuit analysis: impedance Z = √(R² + X²) — for R = 6 Ω and X = 8 Ω, Z = √(36+64) = √100 = 10 Ω, the 6-8-10 triple electricians recognize instantly. With R = 5 and X = 5, Z = √50 = 5√2 ≈ 7.07 Ω — the simplified radical is the exact answer the phasor diagram demands, while 7.07 is the multimeter’s approximation. Power-factor correction and resonance tuning (where inductive and capacitive reactances cancel the radical entirely) all run on simplified radicals. The widget’s factoring chain is the same steps an engineer performs before reaching for the calculator: simplify symbolically, evaluate numerically, trust the symbol.
Polling margins: the standard error √(p(1−p)/n) rarely simplifies in practice, but textbook problems engineer friendly radicands — √(72/100) = √72/10 = 6√2/10 = 3√2/5 ≈ 0.8485. The simplification habit matters because the structure survives: a student who sees 3√2/5 recognizes the √2 from the 45-45-90 triangle and the 3/5 from the 3-4-5 triple, while 0.8485 is just digits. Exact forms compose — multiply two margins and the radicals combine — while decimals merely accumulate rounding. The widget teaches the form that scales.
Optics design: the half-angle formula sin(θ/2) = √((1−cosθ)/2) produces nested radicals — for θ = 45°, sin(22.5°) = √(2−√2)/2, a radical inside a radical. Simplification works layer by layer: the inner √2 is already simplest, and the outer structure stands as the answer. Lens designers meet these in numerical-aperture calculations (NA = n·sinθ), where exact forms verify ray-trace software — if the code’s 0.382683 doesn’t match √(2−√2)/2, the code, not the radical, is wrong. The widget’s single-level simplification is the foundation skill; optics stacks it.
Simplify radicals: factor radicand, extract largest square factor: sqrt(a^2 * b) = a*sqrt(b). Only like radicals add. Source: Sigma Calculator, Simplifying Radicals Calculator, https://sigmacalculator.net/math-calculators/simplifying-radicals-calculator/
Cite this page as: Simplifying Radicals Calculator, Sigma Calculator, https://sigmacalculator.net/math-calculators/simplifying-radicals-calculator/
Last updated: October 12, 2026