Square Root of 215

√215 at a glance

Exact value

√215

Decimal (10 places)

14.6628782986

Rounded

14.7 · 14.66 · 14.663

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.662878

Prime factorization

5 × 43

Cube root

5.990726

How to simplify √215

The prime factorization of 215 is 5 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √215 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 215, 5 and 43 appear an odd number of times, so √215 is irrational and 14.6628782986 is a rounded value.

Where √215 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √215 lies between 14 and 15. 215 is 19 above 196 and 10 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.6552 (0.05% low)
• Tangent from 14, i.e. 14 + 19 ÷ 28: 14.6786 (0.11% high)
• Tangent from 15, i.e. 15 − 10 ÷ 30: 14.6667 (0.03% high)

For √215 the tangent at 15 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 215 is just 10 below 225.

Finding √215 with the Babylonian method

If a guess is too big, 215 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√215) in one step.

Start from the nearest whole number, 15 (15² = 225):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √215 = 14.6628782986 to every decimal shown.

√215 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √215 the pattern is [14; 1, 1, 1, 28] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √215 is irrational.

Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:

The same fractions solve Pell’s equation, x² − 215y² = 1. Its smallest solution in positive whole numbers is x = 44, y = 3.

√215 in geometry and everyday measurements

• A square patio or deck of 215 square feet is about 14.66 ft (14 ft 8 in) on each side, so edging all the way around takes 4 × √215 ≈ 58.7 ft.
• 215 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √215 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √215 as its space diagonal.

Square roots near √215 and related roots

• The cube root of 215 is about 5.990726.
• Four times the radicand doubles the root: √860 = 2 × √215 ≈ 29.325757.

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