Square Root of 291

√291 at a glance

Exact value

√291

Decimal (10 places)

17.0587221092

Rounded

17.1 · 17.06 · 17.059

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.058722

Prime factorization

3 × 97

Cube root

6.626705

How to simplify √291

The prime factorization of 291 is 3 × 97. Every prime appears only once, so there is no pair to bring outside the radical — √291 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 291, 3 and 97 appear an odd number of times, so √291 is irrational and 17.0587221092 is a rounded value.

Where √291 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √291 lies between 17 and 18. 291 is 2 above 289 and 33 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.0571 (0.01% low)
• Tangent from 17, i.e. 17 + 2 ÷ 34: 17.0588 (0% high)
• Tangent from 18, i.e. 18 − 33 ÷ 36: 17.0833 (0.14% high)

For √291 the tangent at 17 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 291 is just 2 above 289.

Finding √291 with the Babylonian method

If a guess is too big, 291 ÷ 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√291) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √291 = 17.0587221092 to every decimal shown.

√291 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √291 the pattern is [17; 17, 34] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √291 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 − 291y² = 1. Its smallest solution in positive whole numbers is x = 290, y = 17.

√291 in geometry and everyday measurements

• A square patio or deck of 291 square feet is about 17.06 ft (17 ft 1 in) on each side, so edging all the way around takes 4 × √291 ≈ 68.2 ft.
• 291 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √291 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 17 box, because 1² + 1² + 17² = 291.

Square roots near √291 and related roots

• The cube root of 291 is about 6.626705.
• Squaring undoes the root: (√291)² = 291, while 291² = 84,681 — the number whose square root is 291.

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