Square Root of 106

√106 at a glance

Exact value

√106

Decimal (10 places)

10.2956301410

Rounded

10.3 · 10.30 · 10.296

Perfect square?

No — between 10² and 11²

Rational?

Irrational

Both square roots

±10.295630

Prime factorization

2 × 53

Cube root

4.732623

How to simplify √106

The prime factorization of 106 is 2 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √106 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 106, 2 and 53 appear an odd number of times, so √106 is irrational and 10.2956301410 is a rounded value.

Where √106 sits between perfect squares

100 = 10² and 121 = 11² are the nearest perfect squares, so √106 lies between 10 and 11. 106 is 6 above 100 and 15 below 121, so the root is closer to 10.

• Straight line between 100 and 121: 10.2857 (0.1% low)
• Tangent from 10, i.e. 10 + 6 ÷ 20: 10.3000 (0.04% high)
• Tangent from 11, i.e. 11 − 15 ÷ 22: 10.3182 (0.22% high)

For √106 the tangent at 10 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 106 is just 6 above 100.

Finding √106 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, 10 (10² = 100):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √106 = 10.2956301410 to every decimal shown.

√106 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √106 the pattern is [10; 3, 2, 1, 1, 1, 1, 2, 3, 20] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √106 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 − 106y² = 1. Its smallest solution in positive whole numbers is x = 32,080,051, y = 3,115,890. Because the period is odd, the equation with −1 on the right also has a solution: 4,005² − 106 × 389² = −1.

√106 in geometry and everyday measurements

• A square room or garden bed covering 106 square feet measures about 10.3 ft (10 ft 4 in) along each wall.
• 106 = 5² + 9², so by the Pythagorean theorem √106 is the diagonal of a 5 × 9 rectangle — and the distance between the points (0, 0) and (5, 9) on a grid.

Square roots near √106 and related roots

• The cube root of 106 is about 4.732623.
• Four times the radicand doubles the root: √424 = 2 × √106 ≈ 20.59126.

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