Square Root of 721

√721 at a glance

Exact value

√721

Decimal (10 places)

26.8514431642

Rounded

26.9 · 26.85 · 26.851

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.851443

Prime factorization

7 × 103

Cube root

8.966957

How to simplify √721

The prime factorization of 721 is 7 × 103. Every prime appears only once, so there is no pair to bring outside the radical — √721 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 721, 7 and 103 appear an odd number of times, so √721 is irrational and 26.8514431642 is a rounded value.

Where √721 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √721 lies between 26 and 27. 721 is 45 above 676 and 8 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.8491 (0.01% low)
• Tangent from 26, i.e. 26 + 45 ÷ 52: 26.8654 (0.05% high)
• Tangent from 27, i.e. 27 − 8 ÷ 54: 26.8519 (0% high)

For √721 the tangent at 27 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 721 is just 8 below 729.

Finding √721 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 721: following the tangent line down to zero simplifies to averaging x with 721 ÷ x.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √721 = 26.8514431642 to every decimal shown.

√721 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √721 the pattern is [26; 1, 5, 1, 2, 1, 2, 1, 1, 1, 1, 1, 1, …] with the block of 40 terms after the semicolon repeating forever (only the first 12 of the 40 are shown). A pattern that never ends is one more proof that √721 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 − 721y² = 1. Its smallest solution in positive whole numbers is x = 18,632,176,943,292,415, y = 693,898,530,122,112 — 17 digits for x, even though 721 is small, which is what makes Pell’s equation famous.

√721 in geometry and everyday measurements

• 721 square feet is 67 m². Laid out as a square — a small house footprint or a lot — it is about 26.85 ft (26 ft 10 in) on a side.
• 721 is not a sum of two whole-number squares — the prime factor 7 and 103 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √721 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 12 × 24 box, because 1² + 12² + 24² = 721.

Square roots near √721 and related roots

• The cube root of 721 is about 8.966957.
• Squaring undoes the root: (√721)² = 721, while 721² = 519,841 — the number whose square root is 721.

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