Square Root of 109

√109 at a glance

Exact value

√109

Decimal (10 places)

10.4403065089

Rounded

10.4 · 10.44 · 10.440

Perfect square?

No — between 10² and 11²

Rational?

Irrational

Both square roots

±10.440307

Prime factorization

109

Cube root

4.776856

How to simplify √109

109 is a prime number, so its only factors are 1 and 109. There is no perfect-square factor to pull out, which means √109 is already in its simplest radical form.

The square root of any prime is irrational. If √109 were a fraction a/b in lowest terms, then a² = 109b², so 109 would divide a — and then 109 would divide b too, contradicting “lowest terms.” That is why the decimal 10.4403065089 is only a rounded value.

Where √109 sits between perfect squares

100 = 10² and 121 = 11² are the nearest perfect squares, so √109 lies between 10 and 11. 109 is 9 above 100 and 12 below 121, so the root is closer to 10.

• Straight line between 100 and 121: 10.4286 (0.11% low)
• Tangent from 10, i.e. 10 + 9 ÷ 20: 10.4500 (0.09% high)
• Tangent from 11, i.e. 11 − 12 ÷ 22: 10.4545 (0.14% high)

For √109 the tangent at 10 wins, missing by only 0.0097. Tangent estimates shine when the number sits close to a perfect square — here 109 is just 9 above 100.

Finding √109 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 109: following the tangent line down to zero simplifies to averaging x with 109 ÷ x.

Start from the nearest whole number, 10 (10² = 100):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √109 = 10.4403065089 to every decimal shown.

√109 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √109 the pattern is [10; 2, 3, 1, 2, 4, 1, 6, 6, 1, 4, 2, 1, …] with the block of 15 terms after the semicolon repeating forever (only the first 12 of the 15 are shown). A pattern that never ends is one more proof that √109 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 − 109y² = 1. Its smallest solution in positive whole numbers is x = 158,070,671,986,249, y = 15,140,424,455,100 — 15 digits for x, even though 109 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: 8,890,182² − 109 × 851,525² = −1.

√109 in geometry and everyday measurements

• A square room or garden bed covering 109 square feet measures about 10.44 ft (10 ft 5 in) along each wall.
• 109 = 3² + 10², so by the Pythagorean theorem √109 is the diagonal of a 3 × 10 rectangle — and the distance between the points (0, 0) and (3, 10) on a grid.

Square roots near √109 and related roots

• The cube root of 109 is about 4.776856.
• Four times the radicand doubles the root: √436 = 2 × √109 ≈ 20.880613.

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