Square Root of 149

√149 at a glance

Exact value

√149

Decimal (10 places)

12.2065556157

Rounded

12.2 · 12.21 · 12.207

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.206556

Prime factorization

149

Cube root

5.301459

How to simplify √149

149 is a prime number, so its only factors are 1 and 149. There is no perfect-square factor to pull out, which means √149 is already in its simplest radical form.

The square root of any prime is irrational. If √149 were a fraction a/b in lowest terms, then a² = 149b², so 149 would divide a — and then 149 would divide b too, contradicting “lowest terms.” That is why the decimal 12.2065556157 is only a rounded value.

Where √149 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √149 lies between 12 and 13. 149 is 5 above 144 and 20 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.2000 (0.05% low)
• Tangent from 12, i.e. 12 + 5 ÷ 24: 12.2083 (0.01% high)
• Tangent from 13, i.e. 13 − 20 ÷ 26: 12.2308 (0.2% high)

For √149 the tangent at 12 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 149 is just 5 above 144.

Finding √149 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 149: following the tangent line down to zero simplifies to averaging x with 149 ÷ x.

Start from the nearest whole number, 12 (12² = 144):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √149 = 12.2065556157 to every decimal shown.

√149 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √149 the pattern is [12; 4, 1, 5, 3, 3, 5, 1, 4, 24] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √149 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 − 149y² = 1. Its smallest solution in positive whole numbers is x = 25,801,741,449, y = 2,113,761,020. Because the period is odd, the equation with −1 on the right also has a solution: 113,582² − 149 × 9,305² = −1.

√149 in geometry and everyday measurements

• A square room or garden bed covering 149 square feet measures about 12.21 ft (12 ft 2 in) along each wall.
• 149 = 7² + 10², so by the Pythagorean theorem √149 is the diagonal of a 7 × 10 rectangle — and the distance between the points (0, 0) and (7, 10) on a grid.

Square roots near √149 and related roots

• The cube root of 149 is about 5.301459.
• Four times the radicand doubles the root: √596 = 2 × √149 ≈ 24.413111.

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