Square Root of 731

√731 at a glance

Exact value

√731

Decimal (10 places)

27.0370116692

Rounded

27.0 · 27.04 · 27.037

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.037012

Prime factorization

17 × 43

Cube root

9.008223

How to simplify √731

The prime factorization of 731 is 17 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √731 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 731, 17 and 43 appear an odd number of times, so √731 is irrational and 27.0370116692 is a rounded value.

Where √731 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √731 lies between 27 and 28. 731 is 2 above 729 and 53 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.0364 (0% low)
• Tangent from 27, i.e. 27 + 2 ÷ 54: 27.0370 (0% high)
• Tangent from 28, i.e. 28 − 53 ÷ 56: 27.0536 (0.06% high)

For √731 the tangent at 27 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 731 is just 2 above 729.

Finding √731 with the Babylonian method

If a guess is too big, 731 ÷ 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√731) in one step.

Start from the nearest whole number, 27 (27² = 729):

Because the starting guess was already close, two steps are enough to match √731 = 27.0370116692 to every decimal shown.

√731 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √731 the pattern is [27; 27, 54] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √731 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 − 731y² = 1. Its smallest solution in positive whole numbers is x = 730, y = 27.

√731 in geometry and everyday measurements

• 731 square feet is 67.9 m². Laid out as a square — a small house footprint or a lot — it is about 27.04 ft (27 ft) on a side.
• 731 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 √731 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 27 box, because 1² + 1² + 27² = 731.

Square roots near √731 and related roots

• The cube root of 731 is about 9.008223.
• Squaring undoes the root: (√731)² = 731, while 731² = 534,361 — the number whose square root is 731.

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