Square Root of 909

√909 at a glance

Exact value

3√101

Decimal (10 places)

30.1496268634

Rounded

30.1 · 30.15 · 30.150

Perfect square?

No — between 30² and 31²

Rational?

Irrational

Both square roots

±30.149627

Prime factorization

3² × 101

Cube root

9.686970

How to simplify √909

Look for the largest perfect square that divides 909. Here it is 9 (3²), because 909 = 9 × 101 and 101 has no square factor left:

The prime factorization tells the same story: 909 = 3² × 101. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 101 stays inside.

Check: (3√101)² = 3² × 101 = 9 × 101 = 909. As a decimal, 3√101 = 3 × 10.0498756211 ≈ 30.1496268634.

Where √909 sits between perfect squares

900 = 30² and 961 = 31² are the nearest perfect squares, so √909 lies between 30 and 31. 909 is 9 above 900 and 52 below 961, so the root is closer to 30.

• Straight line between 900 and 961: 30.1475 (0.01% low)
• Tangent from 30, i.e. 30 + 9 ÷ 60: 30.1500 (0% high)
• Tangent from 31, i.e. 31 − 52 ÷ 62: 30.1613 (0.04% high)

For √909 the tangent at 30 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 909 is just 9 above 900.

Finding √909 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 909: following the tangent line down to zero simplifies to averaging x with 909 ÷ x.

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 √909 = 30.1496268634 to every decimal shown.

√909 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √909 the pattern is [30; 6, 1, 2, 6, 2, 1, 6, 60] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √909 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 − 909y² = 1. Its smallest solution in positive whole numbers is x = 80,801, y = 2,680.

√909 in geometry and everyday measurements

• 909 square feet is 84.4 m². Laid out as a square — a small house footprint or a lot — it is about 30.15 ft (30 ft 2 in) on a side.
• 909 = 3² + 30², so by the Pythagorean theorem √909 is the diagonal of a 3 × 30 rectangle — and the distance between the points (0, 0) and (3, 30) on a grid.
• Since √909 = 3√101, a length of √909 is exactly 3 copies of the length √101 laid end to end.

Square roots near √909 and related roots

• The cube root of 909 is about 9.686970.
• Squaring undoes the root: (√909)² = 909, while 909² = 826,281 — the number whose square root is 909.

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