Square Root of 195

√195 at a glance

Exact value

√195

Decimal (10 places)

13.9642400438

Rounded

14.0 · 13.96 · 13.964

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.964240

Prime factorization

3 × 5 × 13

Cube root

5.798890

How to simplify √195

The prime factorization of 195 is 3 × 5 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √195 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 195, 3, 5 and 13 appear an odd number of times, so √195 is irrational and 13.9642400438 is a rounded value.

Where √195 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √195 lies between 13 and 14. 195 is 26 above 169 and 1 below 196, so the root is closer to 14.

• Straight line between 169 and 196: 13.9630 (0.01% low)
• Tangent from 13, i.e. 13 + 26 ÷ 26: 14.0000 (0.26% high)
• Tangent from 14, i.e. 14 − 1 ÷ 28: 13.9643 (0% high)

For √195 the tangent at 14 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 195 is just 1 below 196.

Finding √195 with the Babylonian method

If a guess is too big, 195 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√195) in one step.

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 √195 = 13.9642400438 to every decimal shown.

√195 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √195 the pattern is [13; 1, 26] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √195 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 − 195y² = 1. Its smallest solution in positive whole numbers is x = 14, y = 1.

√195 in geometry and everyday measurements

• A square room or garden bed covering 195 square feet measures about 13.96 ft (14 ft) along each wall.
• 195 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √195 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 13 box, because 1² + 5² + 13² = 195.

Square roots near √195 and related roots

• The cube root of 195 is about 5.798890.
• Four times the radicand doubles the root: √780 = 2 × √195 ≈ 27.92848.

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