Square Root of 131

√131 at a glance

Exact value

√131

Decimal (10 places)

11.4455231423

Rounded

11.4 · 11.45 · 11.446

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.445523

Prime factorization

131

Cube root

5.078753

How to simplify √131

131 is a prime number, so its only factors are 1 and 131. There is no perfect-square factor to pull out, which means √131 is already in its simplest radical form.

The square root of any prime is irrational. If √131 were a fraction a/b in lowest terms, then a² = 131b², so 131 would divide a — and then 131 would divide b too, contradicting “lowest terms.” That is why the decimal 11.4455231423 is only a rounded value.

Where √131 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √131 lies between 11 and 12. 131 is 10 above 121 and 13 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.4348 (0.09% low)
• Tangent from 11, i.e. 11 + 10 ÷ 22: 11.4545 (0.08% high)
• Tangent from 12, i.e. 12 − 13 ÷ 24: 11.4583 (0.11% high)

For √131 the tangent at 11 wins, missing by only 0.009. Tangent estimates shine when the number sits close to a perfect square — here 131 is just 10 above 121.

Finding √131 with the Babylonian method

If a guess is too big, 131 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√131) in one step.

Start from the nearest whole number, 11 (11² = 121):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √131 = 11.4455231423 to every decimal shown.

√131 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √131 the pattern is [11; 2, 4, 11, 4, 2, 22] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √131 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 − 131y² = 1. Its smallest solution in positive whole numbers is x = 10,610, y = 927.

√131 in geometry and everyday measurements

• A square room or garden bed covering 131 square feet measures about 11.45 ft (11 ft 5 in) along each wall.
• 131 is not a sum of two whole-number squares — 131 is itself a prime that is one less than a multiple of 4, which rules that out — so √131 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 11 box, because 1² + 3² + 11² = 131.

Square roots near √131 and related roots

• The cube root of 131 is about 5.078753.
• Four times the radicand doubles the root: √524 = 2 × √131 ≈ 22.891046.

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