Square Root of 197

√197 at a glance

Exact value

√197

Decimal (10 places)

14.0356688476

Rounded

14.0 · 14.04 · 14.036

Perfect square?

No — between 14² and 15²

Rational?

Irrational

Both square roots

±14.035669

Prime factorization

197

Cube root

5.818648

How to simplify √197

197 is a prime number, so its only factors are 1 and 197. There is no perfect-square factor to pull out, which means √197 is already in its simplest radical form.

The square root of any prime is irrational. If √197 were a fraction a/b in lowest terms, then a² = 197b², so 197 would divide a — and then 197 would divide b too, contradicting “lowest terms.” That is why the decimal 14.0356688476 is only a rounded value.

Where √197 sits between perfect squares

196 = 14² and 225 = 15² are the nearest perfect squares, so √197 lies between 14 and 15. 197 is 1 above 196 and 28 below 225, so the root is closer to 14.

• Straight line between 196 and 225: 14.0345 (0.01% low)
• Tangent from 14, i.e. 14 + 1 ÷ 28: 14.0357 (0% high)
• Tangent from 15, i.e. 15 − 28 ÷ 30: 14.0667 (0.22% high)

For √197 the tangent at 14 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 197 is just 1 above 196.

Finding √197 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 197: following the tangent line down to zero simplifies to averaging x with 197 ÷ x.

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

The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √197 = 14.0356688476 to every decimal shown.

√197 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √197 the pattern is [14; 28] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 197 is one more than a perfect square (14² + 1). A pattern that never ends is one more proof that √197 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 − 197y² = 1. Its smallest solution in positive whole numbers is x = 393, y = 28. Because the period is odd, the equation with −1 on the right also has a solution: 14² − 197 × 1² = −1.

√197 in geometry and everyday measurements

• A square room or garden bed covering 197 square feet measures about 14.04 ft (14 ft) along each wall.
• 197 = 1² + 14², so by the Pythagorean theorem √197 is the diagonal of a 1 × 14 rectangle — and the distance between the points (0, 0) and (1, 14) on a grid.

Square roots near √197 and related roots

• The cube root of 197 is about 5.818648.
• Four times the radicand doubles the root: √788 = 2 × √197 ≈ 28.071338.

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