Square Root of 808

√808 at a glance

Exact value

2√202

Decimal (10 places)

28.4253408071

Rounded

28.4 · 28.43 · 28.425

Perfect square?

No — between 28² and 29²

Rational?

Irrational

Both square roots

±28.425341

Prime factorization

2³ × 101

Cube root

9.314019

How to simplify √808

Look for the largest perfect square that divides 808. Here it is 4 (2²), because 808 = 4 × 202 and 202 has no square factor left:

The prime factorization tells the same story: 808 = 2³ × 101. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 101 stays inside.

Check: (2√202)² = 2² × 202 = 4 × 202 = 808. As a decimal, 2√202 = 2 × 14.2126704036 ≈ 28.4253408071.

Where √808 sits between perfect squares

784 = 28² and 841 = 29² are the nearest perfect squares, so √808 lies between 28 and 29. 808 is 24 above 784 and 33 below 841, so the root is closer to 28.

• Straight line between 784 and 841: 28.4211 (0.02% low)
• Tangent from 28, i.e. 28 + 24 ÷ 56: 28.4286 (0.01% high)
• Tangent from 29, i.e. 29 − 33 ÷ 58: 28.4310 (0.02% high)

For √808 the tangent at 28 wins, missing by only 0.0032. Tangent estimates shine when the number sits close to a perfect square — here 808 is just 24 above 784.

Finding √808 with the Babylonian method

Picture a rectangle with an area of 808 and one side x; the other side must be 808 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √808.

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 √808 = 28.4253408071 to every decimal shown.

√808 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √808 the pattern is [28; 2, 2, 1, 5, 1, 1, 1, 1, 13, 1, 1, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √808 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 − 808y² = 1. Its smallest solution in positive whole numbers is x = 19,731,763, y = 694,161.

√808 in geometry and everyday measurements

• 808 square feet is 75.1 m². Laid out as a square — a small house footprint or a lot — it is about 28.43 ft (28 ft 5 in) on a side.
• 808 = 18² + 22², so by the Pythagorean theorem √808 is the diagonal of a 18 × 22 rectangle — and the distance between the points (0, 0) and (18, 22) on a grid.
• Since √808 = 2√202, a length of √808 is exactly 2 copies of the length √202 laid end to end.

Square roots near √808 and related roots

• The cube root of 808 is about 9.314019.
• Because 808 = 4 × 202, the root is twice √202: 2 × 14.21267 ≈ 28.425341.

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