Square Root of 299

√299 at a glance

Exact value

√299

Decimal (10 places)

17.2916164658

Rounded

17.3 · 17.29 · 17.292

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.291616

Prime factorization

13 × 23

Cube root

6.686883

How to simplify √299

The prime factorization of 299 is 13 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √299 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 299, 13 and 23 appear an odd number of times, so √299 is irrational and 17.2916164658 is a rounded value.

Where √299 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √299 lies between 17 and 18. 299 is 10 above 289 and 25 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.2857 (0.03% low)
• Tangent from 17, i.e. 17 + 10 ÷ 34: 17.2941 (0.01% high)
• Tangent from 18, i.e. 18 − 25 ÷ 36: 17.3056 (0.08% high)

For √299 the tangent at 17 wins, missing by only 0.0025. Tangent estimates shine when the number sits close to a perfect square — here 299 is just 10 above 289.

Finding √299 with the Babylonian method

If a guess is too big, 299 ÷ 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√299) in one step.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √299 = 17.2916164658 to every decimal shown.

√299 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √299 the pattern is [17; 3, 2, 3, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √299 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 − 299y² = 1. Its smallest solution in positive whole numbers is x = 415, y = 24.

√299 in geometry and everyday measurements

• A square patio or deck of 299 square feet is about 17.29 ft (17 ft 3 in) on each side, so edging all the way around takes 4 × √299 ≈ 69.2 ft.
• 299 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √299 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 17 box, because 1² + 3² + 17² = 299.

Square roots near √299 and related roots

• The cube root of 299 is about 6.686883.
• Squaring undoes the root: (√299)² = 299, while 299² = 89,401 — the number whose square root is 299.

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