Square Root of 723

√723 at a glance

Exact value

√723

Decimal (10 places)

26.8886593195

Rounded

26.9 · 26.89 · 26.889

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.888659

Prime factorization

3 × 241

Cube root

8.975241

How to simplify √723

The prime factorization of 723 is 3 × 241. Every prime appears only once, so there is no pair to bring outside the radical — √723 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 723, 3 and 241 appear an odd number of times, so √723 is irrational and 26.8886593195 is a rounded value.

Where √723 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √723 lies between 26 and 27. 723 is 47 above 676 and 6 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.8868 (0.01% low)
• Tangent from 26, i.e. 26 + 47 ÷ 52: 26.9038 (0.06% high)
• Tangent from 27, i.e. 27 − 6 ÷ 54: 26.8889 (0% high)

For √723 the tangent at 27 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 723 is just 6 below 729.

Finding √723 with the Babylonian method

If a guess is too big, 723 ÷ 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√723) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √723 = 26.8886593195 to every decimal shown.

√723 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √723 the pattern is [26; 1, 7, 1, 52] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √723 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 − 723y² = 1. Its smallest solution in positive whole numbers is x = 242, y = 9.

√723 in geometry and everyday measurements

• 723 square feet is 67.2 m². Laid out as a square — a small house footprint or a lot — it is about 26.89 ft (26 ft 11 in) on a side.
• 723 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 √723 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 19 × 19 box, because 1² + 19² + 19² = 723.

Square roots near √723 and related roots

• The cube root of 723 is about 8.975241.
• Squaring undoes the root: (√723)² = 723, while 723² = 522,729 — the number whose square root is 723.

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