Square Root of 809

√809 at a glance

Exact value

√809

Decimal (10 places)

28.4429253067

Rounded

28.4 · 28.44 · 28.443

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.442925

Prime factorization

809

Cube root

9.317860

How to simplify √809

809 is a prime number, so its only factors are 1 and 809. There is no perfect-square factor to pull out, which means √809 is already in its simplest radical form.

The square root of any prime is irrational. If √809 were a fraction a/b in lowest terms, then a² = 809b², so 809 would divide a — and then 809 would divide b too, contradicting “lowest terms.” That is why the decimal 28.4429253067 is only a rounded value.

Where √809 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √809 lies between 28 and 29. 809 is 25 above 784 and 32 below 841, so the root is closer to 28.

• Straight line between 784 and 841: 28.4386 (0.02% low)
• Tangent from 28, i.e. 28 + 25 ÷ 56: 28.4464 (0.01% high)
• Tangent from 29, i.e. 29 − 32 ÷ 58: 28.4483 (0.02% high)

For √809 the tangent at 28 wins, missing by only 0.0035. Tangent estimates shine when the number sits close to a perfect square — here 809 is just 25 above 784.

Finding √809 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 809: following the tangent line down to zero simplifies to averaging x with 809 ÷ x.

Start from the nearest whole number, 28 (28² = 784):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √809 = 28.4429253067 to every decimal shown.

√809 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √809 the pattern is [28; 2, 3, 1, 7, 2, 1, 6, 2, 3, 11, 11, 3, …] with the block of 21 terms after the semicolon repeating forever (only the first 12 of the 21 are shown). A pattern that never ends is one more proof that √809 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 − 809y² = 1. Its smallest solution in positive whole numbers is x = 376,455,160,998,025,676,163,201, y = 13,235,458,622,462,202,510,640 — 24 digits for x, even though 809 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 433,852,026,040² − 809 × 15,253,424,933² = −1.

√809 in geometry and everyday measurements

• 809 square feet is 75.2 m². Laid out as a square — a small house footprint or a lot — it is about 28.44 ft (28 ft 5 in) on a side.
• 809 = 5² + 28², so by the Pythagorean theorem √809 is the diagonal of a 5 × 28 rectangle — and the distance between the points (0, 0) and (5, 28) on a grid.

Square roots near √809 and related roots

• The cube root of 809 is about 9.317860.
• Squaring undoes the root: (√809)² = 809, while 809² = 654,481 — the number whose square root is 809.

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