Square Root of 104

√104 at a glance

Exact value

2√26

Decimal (10 places)

10.1980390272

Rounded

10.2 · 10.20 · 10.198

Perfect square?

No — between 10² and 11²

Rational?

Irrational

Both square roots

±10.198039

Prime factorization

2³ × 13

Cube root

4.702669

How to simplify √104

Look for the largest perfect square that divides 104. Here it is 4 (2²), because 104 = 4 × 26 and 26 has no square factor left:

The prime factorization tells the same story: 104 = 2³ × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 13 stays inside.

Check: (2√26)² = 2² × 26 = 4 × 26 = 104. As a decimal, 2√26 = 2 × 5.0990195136 ≈ 10.1980390272.

Where √104 sits between perfect squares

100 = 10² and 121 = 11² are the nearest perfect squares, so √104 lies between 10 and 11. 104 is 4 above 100 and 17 below 121, so the root is closer to 10.

• Straight line between 100 and 121: 10.1905 (0.07% low)
• Tangent from 10, i.e. 10 + 4 ÷ 20: 10.2000 (0.02% high)
• Tangent from 11, i.e. 11 − 17 ÷ 22: 10.2273 (0.29% high)

For √104 the tangent at 10 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 104 is just 4 above 100.

Finding √104 with the Babylonian method

Picture a rectangle with an area of 104 and one side x; the other side must be 104 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √104.

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 √104 = 10.1980390272 to every decimal shown.

√104 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √104 the pattern is [10; 5, 20] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √104 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 − 104y² = 1. Its smallest solution in positive whole numbers is x = 51, y = 5.

√104 in geometry and everyday measurements

• A square room or garden bed covering 104 square feet measures about 10.2 ft (10 ft 2 in) along each wall.
• 104 = 2² + 10², so by the Pythagorean theorem √104 is the diagonal of a 2 × 10 rectangle — and the distance between the points (0, 0) and (2, 10) on a grid.
• Since √104 = 2√26, a length of √104 is exactly 2 copies of the length √26 laid end to end.

Square roots near √104 and related roots

• The cube root of 104 is about 4.702669.
• Four times the radicand doubles the root: √416 = 2 × √104 ≈ 20.396078.

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