Square Root of 908

√908 at a glance

Exact value

2√227

Decimal (10 places)

30.1330383466

Rounded

30.1 · 30.13 · 30.133

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.133038

Prime factorization

2² × 227

Cube root

9.683417

How to simplify √908

Look for the largest perfect square that divides 908. Here it is 4 (2²), because 908 = 4 × 227 and 227 has no square factor left:

The prime factorization tells the same story: 908 = 2² × 227. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 227 stays inside.

Check: (2√227)² = 2² × 227 = 4 × 227 = 908. As a decimal, 2√227 = 2 × 15.0665191733 ≈ 30.1330383466.

Where √908 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √908 lies between 30 and 31. 908 is 8 above 900 and 53 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.1311 (0.01% low)
• Tangent from 30, i.e. 30 + 8 ÷ 60: 30.1333 (0% high)
• Tangent from 31, i.e. 31 − 53 ÷ 62: 30.1452 (0.04% high)

For √908 the tangent at 30 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 908 is just 8 above 900.

Finding √908 with the Babylonian method

Picture a rectangle with an area of 908 and one side x; the other side must be 908 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √908.

Start from the nearest whole number, 30 (30² = 900):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √908 = 30.1330383466 to every decimal shown.

√908 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √908 the pattern is [30; 7, 1, 1, 14, 1, 1, 7, 60] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √908 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 − 908y² = 1. Its smallest solution in positive whole numbers is x = 102,151, y = 3,390.

√908 in geometry and everyday measurements

• 908 square feet is 84.4 m². Laid out as a square — a small house footprint or a lot — it is about 30.13 ft (30 ft 2 in) on a side.
• 908 is not a sum of two whole-number squares — the prime factor 227 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √908 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 30 box, because 2² + 2² + 30² = 908.
• Since √908 = 2√227, a length of √908 is exactly 2 copies of the length √227 laid end to end.

Square roots near √908 and related roots

• The cube root of 908 is about 9.683417.
• Because 908 = 4 × 227, the root is twice √227: 2 × 15.066519 ≈ 30.133038.

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