Square Root of 146

√146 at a glance

Exact value

√146

Decimal (10 places)

12.0830459736

Rounded

12.1 · 12.08 · 12.083

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.083046

Prime factorization

2 × 73

Cube root

5.265637

How to simplify √146

The prime factorization of 146 is 2 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √146 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 146, 2 and 73 appear an odd number of times, so √146 is irrational and 12.0830459736 is a rounded value.

Where √146 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √146 lies between 12 and 13. 146 is 2 above 144 and 23 below 169, so the root is closer to 12.

• Straight line between 144 and 169: 12.0800 (0.03% low)
• Tangent from 12, i.e. 12 + 2 ÷ 24: 12.0833 (0% high)
• Tangent from 13, i.e. 13 − 23 ÷ 26: 12.1154 (0.27% high)

For √146 the tangent at 12 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 146 is just 2 above 144.

Finding √146 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, 12 (12² = 144):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √146 = 12.0830459736 to every decimal shown.

√146 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √146 the pattern is [12; 12, 24] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √146 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 − 146y² = 1. Its smallest solution in positive whole numbers is x = 145, y = 12.

√146 in geometry and everyday measurements

• A square room or garden bed covering 146 square feet measures about 12.08 ft (12 ft 1 in) along each wall.
• 146 = 5² + 11², so by the Pythagorean theorem √146 is the diagonal of a 5 × 11 rectangle — and the distance between the points (0, 0) and (5, 11) on a grid.

Square roots near √146 and related roots

• The cube root of 146 is about 5.265637.
• Four times the radicand doubles the root: √584 = 2 × √146 ≈ 24.166092.

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