√ Radical Simplifier & Surd Calculator
√180 = 6√5 — via the prime factorization, shown. Plus rationalizing c/(a + b√m) with the conjugate, and combining like surds (3√8 + 5√18 = 21√2).
√180 = 6√5
Working
- Prime-factorize: 180 = 2² × 3² × 5
- Pull each pair of primes out of the root (a pair p² contributes p outside): outside = 6, left inside = 5
- √180 = √(6² × 5) = 6√5
Exact symbolic results — √180 becomes 6√5 via prime factorization, never 13.416…. Runs locally.
How the radical simplifier & surd calculator works
Simplifying a radical means pulling every square factor out of the root: the tool prime-factorizes the radicand, extracts each pair of primes (a pair p² contributes p outside the root), and shows exactly that working — √180 = √(6² × 5) = 6√5. The rationalize tab multiplies a fraction like 1/(2 + √3) by the denominator's conjugate, using the difference of squares to clear the root, and presents the exact result with steps. The combine tab simplifies each surd first and then adds, subtracts or multiplies: 3√8 + 5√18 becomes 6√2 + 15√2 = 21√2, while genuinely unlike surds are honestly left as an exact sum rather than mashed into a decimal.
The "simplify each surd FIRST" habit is the whole trick for combining: 3√8 and 5√18 look unlike, but both reduce to multiples of √2. And rationalizing isn't just convention — the exact form (2 − √3) is usable in the next algebra step, where 0.2679… is a dead end. Everything here stays symbolic; decimals are never substituted for answers.
Frequently asked questions
How do I simplify a square root like √180?
Prime-factorize: 180 = 2² × 3² × 5. Each pair leaves the root as a single factor: 2 × 3 = 6 outside, 5 left inside → 6√5. The tool shows this factorization working for any number.
What does it mean to rationalize a denominator?
Rewriting a fraction so no root remains below the line. For 1/(2 + √3), multiply top and bottom by the conjugate (2 − √3): the denominator becomes 2² − 3 = 1, so the result is exactly 2 − √3.
When can two surds be added?
Only when they simplify to the same radicand ("like surds"): 3√8 + 5√18 = 6√2 + 15√2 = 21√2. If the simplified radicands differ (√2 vs √3), the exact sum is already the simplest form — no calculator can merge them without going decimal.
How are surds multiplied?
√a × √b = √(ab), then simplify the product: 2√6 × 3√10 = 6√60 = 6 × 2√15 = 12√15. The tool shows each stage.
Is anything computed with floating point?
No — factorization and coefficient arithmetic are exact (BigInt and rationals). Decimal approximations are never used internally, and everything runs locally.