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√50 = 5√2 ≈ 7.0711. √144 = 12 exactly. ³√27 = 3, ⁴√625 = 5. Decimal and simplified radical, every time.
Math homework wants the exact answer (5√2) and real life wants the decimal (7.0711) — this calculator gives both. It extracts the largest perfect-square factor to simplify radicals automatically, flags perfect squares, and handles cube and nth roots for the full range from geometry homework to engineering quick checks.
Students use it for exact answers: √50 = 5√2 = 7.0711 — simplified radical and decimal together, plus cube and nth roots (³√27 = 3, ⁴√625 = 5).
Factor out the largest perfect square: √50 = √(25 × 2) = √25 × √2 = 5√2. The method: find the biggest perfect square (4, 9, 16, 25, 36, 49, 64, 81, 100…) that divides your number, pull its root outside, leave the remainder inside. √72 = √(36 × 2) = 6√2; √200 = √(100 × 2) = 10√2. A radical is fully simplified when nothing inside has a perfect-square factor — √2, √3, √10 are done; √12 is not (it is 2√3). Why teachers insist: simplified radicals are exact, while 7.0710678… is always a rounding. When the answer feeds another calculation, the radical form keeps the precision; when it feeds a tape measure, the decimal wins.
Bracket it between perfect squares and refine: √50 sits between √49 = 7 and √64 = 8, and since 50 is much closer to 49, estimate just above 7 — say 7.1 (true: 7.0711). The one-step Newton refinement makes it scary accurate: average your guess with number ÷ guess. For √50 with guess 7: (7 + 50/7)/2 = (7 + 7.143)/2 = 7.071 — one step lands within 0.001. Another round nails it to six digits. Worth memorizing: √2 ≈ 1.414, √3 ≈ 1.732, √5 ≈ 2.236, √10 ≈ 3.162 — these four cover most estimation needs, and √2 in particular appears constantly (the diagonal of a square, A4 paper ratios, RMS voltage).
Cube roots reverse volume: a 27 ft³ cube has 3 ft edges, and doubling a cube's volume means edges grow by ³√2 ≈ 1.26, not 2 — the classic scaling insight. Nth roots are everywhere in finance and science: annual growth rates ARE nth roots — an investment that doubles in 5 years grew by ⁵√2 − 1 = 14.87% per year (that is CAGR). Photography f-stops step by √2, music octaves divide into 12 semitones of ¹²√2 ≈ 1.0595 each, and any "average multiplier over n periods" is an nth root. The sign rules: even roots of negatives are not real numbers (no real √−4); odd roots of negatives are fine (³√−27 = −3). This calculator handles positive inputs and flags the rest.