Square Root of 208

√208 at a glance

Exact value

4√13

Decimal (10 places)

14.4222051019

Rounded

14.4 · 14.42 · 14.422

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.422205

Prime factorization

2⁴ × 13

Cube root

5.924992

How to simplify √208

Look for the largest perfect square that divides 208. Here it is 16 (4²), because 208 = 16 × 13 and 13 has no square factor left:

The prime factorization tells the same story: 208 = 2⁴ × 13. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 13 stays inside.

208 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √208 = 2√52, and √52 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√13)² = 4² × 13 = 16 × 13 = 208. As a decimal, 4√13 = 4 × 3.6055512755 ≈ 14.4222051019.

Where √208 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √208 lies between 14 and 15. 208 is 12 above 196 and 17 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.4138 (0.06% low)
• Tangent from 14, i.e. 14 + 12 ÷ 28: 14.4286 (0.04% high)
• Tangent from 15, i.e. 15 − 17 ÷ 30: 14.4333 (0.08% high)

For √208 the tangent at 14 wins, missing by only 0.0064. Tangent estimates shine when the number sits close to a perfect square — here 208 is just 12 above 196.

Finding √208 with the Babylonian method

Picture a rectangle with an area of 208 and one side x; the other side must be 208 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √208.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √208 = 14.4222051019 to every decimal shown.

√208 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √208 the pattern is [14; 2, 2, 1, 2, 2, 28] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √208 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 − 208y² = 1. Its smallest solution in positive whole numbers is x = 649, y = 45.

√208 in geometry and everyday measurements

• A square patio or deck of 208 square feet is about 14.42 ft (14 ft 5 in) on each side, so edging all the way around takes 4 × √208 ≈ 57.7 ft.
• 208 = 8² + 12², so by the Pythagorean theorem √208 is the diagonal of a 8 × 12 rectangle — and the distance between the points (0, 0) and (8, 12) on a grid.
• Since √208 = 4√13, a length of √208 is exactly 4 copies of the length √13 laid end to end.

Square roots near √208 and related roots

• The cube root of 208 is about 5.924992.
• Four times the radicand doubles the root: √832 = 2 × √208 ≈ 28.84441.

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