Square Root of 198

√198 at a glance

Exact value

3√22

Decimal (10 places)

14.0712472795

Rounded

14.1 · 14.07 · 14.071

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.071247

Prime factorization

2 × 3² × 11

Cube root

5.828477

How to simplify √198

Look for the largest perfect square that divides 198. Here it is 9 (3²), because 198 = 9 × 22 and 22 has no square factor left:

The prime factorization tells the same story: 198 = 2 × 3² × 11. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 11 stays inside.

Check: (3√22)² = 3² × 22 = 9 × 22 = 198. As a decimal, 3√22 = 3 × 4.6904157598 ≈ 14.0712472795.

Where √198 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √198 lies between 14 and 15. 198 is 2 above 196 and 27 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.0690 (0.02% low)
• Tangent from 14, i.e. 14 + 2 ÷ 28: 14.0714 (0% high)
• Tangent from 15, i.e. 15 − 27 ÷ 30: 14.1000 (0.2% high)

For √198 the tangent at 14 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 198 is just 2 above 196.

Finding √198 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, 14 (14² = 196):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √198 = 14.0712472795 to every decimal shown.

√198 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √198 the pattern is [14; 14, 28] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √198 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 − 198y² = 1. Its smallest solution in positive whole numbers is x = 197, y = 14.

√198 in geometry and everyday measurements

• A square room or garden bed covering 198 square feet measures about 14.07 ft (14 ft 1 in) along each wall.
• 198 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √198 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 14 box, because 1² + 1² + 14² = 198.
• Since √198 = 3√22, a length of √198 is exactly 3 copies of the length √22 laid end to end.

Square roots near √198 and related roots

• The cube root of 198 is about 5.828477.
• Four times the radicand doubles the root: √792 = 2 × √198 ≈ 28.142495.

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