Square Root of 726

√726 at a glance

Exact value

11√6

Decimal (10 places)

26.9443871706

Rounded

26.9 · 26.94 · 26.944

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.944387

Prime factorization

2 × 3 × 11²

Cube root

8.987637

How to simplify √726

Look for the largest perfect square that divides 726. Here it is 121 (11²), because 726 = 121 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 726 = 2 × 3 × 11². Each pair of equal primes leaves the radical as one factor, so 11 comes out and 2 × 3 stays inside.

Check: (11√6)² = 11² × 6 = 121 × 6 = 726. As a decimal, 11√6 = 11 × 2.4494897428 ≈ 26.9443871706.

Where √726 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √726 lies between 26 and 27. 726 is 50 above 676 and 3 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.9434 (0% low)
• Tangent from 26, i.e. 26 + 50 ÷ 52: 26.9615 (0.06% high)
• Tangent from 27, i.e. 27 − 3 ÷ 54: 26.9444 (0% high)

For √726 the tangent at 27 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 726 is just 3 below 729.

Finding √726 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √726 = 26.9443871706 to every decimal shown.

√726 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √726 the pattern is [26; 1, 16, 1, 52] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √726 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 − 726y² = 1. Its smallest solution in positive whole numbers is x = 485, y = 18.

√726 in geometry and everyday measurements

• 726 square feet is 67.4 m². Laid out as a square — a small house footprint or a lot — it is about 26.94 ft (26 ft 11 in) on a side.
• 726 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 √726 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 26 box, because 1² + 7² + 26² = 726.
• Since √726 = 11√6, a length of √726 is exactly 11 copies of the length √6 laid end to end.

Square roots near √726 and related roots

• The cube root of 726 is about 8.987637.
• Squaring undoes the root: (√726)² = 726, while 726² = 527,076 — the number whose square root is 726.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.