Square Root of 137

√137 at a glance

Exact value

√137

Decimal (10 places)

11.7046999107

Rounded

11.7 · 11.70 · 11.705

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.704700

Prime factorization

137

Cube root

5.155137

How to simplify √137

137 is a prime number, so its only factors are 1 and 137. There is no perfect-square factor to pull out, which means √137 is already in its simplest radical form.

The square root of any prime is irrational. If √137 were a fraction a/b in lowest terms, then a² = 137b², so 137 would divide a — and then 137 would divide b too, contradicting “lowest terms.” That is why the decimal 11.7046999107 is only a rounded value.

Where √137 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √137 lies between 11 and 12. 137 is 16 above 121 and 7 below 144, so the root is closer to 12.

• Straight line between 121 and 144: 11.6957 (0.08% low)
• Tangent from 11, i.e. 11 + 16 ÷ 22: 11.7273 (0.19% high)
• Tangent from 12, i.e. 12 − 7 ÷ 24: 11.7083 (0.03% high)

For √137 the tangent at 12 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 137 is just 7 below 144.

Finding √137 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 137: following the tangent line down to zero simplifies to averaging x with 137 ÷ 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 √137 = 11.7046999107 to every decimal shown.

√137 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √137 the pattern is [11; 1, 2, 2, 1, 1, 2, 2, 1, 22] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √137 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 − 137y² = 1. Its smallest solution in positive whole numbers is x = 6,083,073, y = 519,712. Because the period is odd, the equation with −1 on the right also has a solution: 1,744² − 137 × 149² = −1.

√137 in geometry and everyday measurements

• A square room or garden bed covering 137 square feet measures about 11.7 ft (11 ft 8 in) along each wall.
• 137 = 4² + 11², so by the Pythagorean theorem √137 is the diagonal of a 4 × 11 rectangle — and the distance between the points (0, 0) and (4, 11) on a grid.

Square roots near √137 and related roots

• The cube root of 137 is about 5.155137.
• Four times the radicand doubles the root: √548 = 2 × √137 ≈ 23.4094.

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