Square Root of 217

√217 at a glance

Exact value

√217

Decimal (10 places)

14.7309198627

Rounded

14.7 · 14.73 · 14.731

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.730920

Prime factorization

7 × 31

Cube root

6.009245

How to simplify √217

The prime factorization of 217 is 7 × 31. Every prime appears only once, so there is no pair to bring outside the radical — √217 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 217, 7 and 31 appear an odd number of times, so √217 is irrational and 14.7309198627 is a rounded value.

Where √217 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √217 lies between 14 and 15. 217 is 21 above 196 and 8 below 225, so the root is closer to 15.

• Straight line between 196 and 225: 14.7241 (0.05% low)
• Tangent from 14, i.e. 14 + 21 ÷ 28: 14.7500 (0.13% high)
• Tangent from 15, i.e. 15 − 8 ÷ 30: 14.7333 (0.02% high)

For √217 the tangent at 15 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 217 is just 8 below 225.

Finding √217 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 217: following the tangent line down to zero simplifies to averaging x with 217 ÷ x.

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 √217 = 14.7309198627 to every decimal shown.

√217 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √217 the pattern is [14; 1, 2, 1, 2, 1, 1, 9, 4, 9, 1, 1, 2, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √217 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 − 217y² = 1. Its smallest solution in positive whole numbers is x = 3,844,063, y = 260,952.

√217 in geometry and everyday measurements

• A square patio or deck of 217 square feet is about 14.73 ft (14 ft 9 in) on each side, so edging all the way around takes 4 × √217 ≈ 58.9 ft.
• 217 is not a sum of two whole-number squares — the prime factor 7 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √217 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 8 × 12 box, because 3² + 8² + 12² = 217.

Square roots near √217 and related roots

• The cube root of 217 is about 6.009245.
• Four times the radicand doubles the root: √868 = 2 × √217 ≈ 29.46184.

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