Square Root of 130

√130 at a glance

Exact value

√130

Decimal (10 places)

11.4017542510

Rounded

11.4 · 11.40 · 11.402

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.401754

Prime factorization

2 × 5 × 13

Cube root

5.065797

How to simplify √130

The prime factorization of 130 is 2 × 5 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √130 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 130, 2, 5 and 13 appear an odd number of times, so √130 is irrational and 11.4017542510 is a rounded value.

Where √130 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √130 lies between 11 and 12. 130 is 9 above 121 and 14 below 144, so the root is closer to 11.

• Straight line between 121 and 144: 11.3913 (0.09% low)
• Tangent from 11, i.e. 11 + 9 ÷ 22: 11.4091 (0.06% high)
• Tangent from 12, i.e. 12 − 14 ÷ 24: 11.4167 (0.13% high)

For √130 the tangent at 11 wins, missing by only 0.0073. Tangent estimates shine when the number sits close to a perfect square — here 130 is just 9 above 121.

Finding √130 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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 √130 = 11.4017542510 to every decimal shown.

√130 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √130 the pattern is [11; 2, 2, 22] with the block of 3 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √130 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 − 130y² = 1. Its smallest solution in positive whole numbers is x = 6,499, y = 570. Because the period is odd, the equation with −1 on the right also has a solution: 57² − 130 × 5² = −1.

√130 in geometry and everyday measurements

• A square room or garden bed covering 130 square feet measures about 11.4 ft (11 ft 5 in) along each wall.
• 130 = 3² + 11² = 7² + 9², so by the Pythagorean theorem √130 is the diagonal of rectangles measuring 3 × 11 and 7 × 9 — and the distance between the points (0, 0) and (3, 11) on a grid.

Square roots near √130 and related roots

• The cube root of 130 is about 5.065797.
• Four times the radicand doubles the root: √520 = 2 × √130 ≈ 22.803509.

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