Square Root of 699

√699 at a glance

Exact value

√699

Decimal (10 places)

26.4386081328

Rounded

26.4 · 26.44 · 26.439

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.438608

Prime factorization

3 × 233

Cube root

8.874810

How to simplify √699

The prime factorization of 699 is 3 × 233. Every prime appears only once, so there is no pair to bring outside the radical — √699 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 699, 3 and 233 appear an odd number of times, so √699 is irrational and 26.4386081328 is a rounded value.

Where √699 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √699 lies between 26 and 27. 699 is 23 above 676 and 30 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.4340 (0.02% low)
• Tangent from 26, i.e. 26 + 23 ÷ 52: 26.4423 (0.01% high)
• Tangent from 27, i.e. 27 − 30 ÷ 54: 26.4444 (0.02% high)

For √699 the tangent at 26 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 699 is just 23 above 676.

Finding √699 with the Babylonian method

If a guess is too big, 699 ÷ 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√699) in one step.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √699 = 26.4386081328 to every decimal shown.

√699 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √699 the pattern is [26; 2, 3, 1, 1, 2, 1, 25, 1, 2, 1, 1, 3, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √699 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 − 699y² = 1. Its smallest solution in positive whole numbers is x = 2,271,050, y = 85,899.

√699 in geometry and everyday measurements

• 699 square feet is 64.9 m². Laid out as a square — a small house footprint or a lot — it is about 26.44 ft (26 ft 5 in) on a side.
• 699 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 √699 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 13 × 23 box, because 1² + 13² + 23² = 699.

Square roots near √699 and related roots

• The cube root of 699 is about 8.874810.
• Squaring undoes the root: (√699)² = 699, while 699² = 488,601 — the number whose square root is 699.

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