Square Root of 39

√39 at a glance

Exact value

√39

Decimal (10 places)

6.2449979984

Rounded

6.2 · 6.24 · 6.245

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.244998

Prime factorization

3 × 13

Cube root

3.391211

How to simplify √39

The prime factorization of 39 is 3 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √39 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 39, 3 and 13 appear an odd number of times, so √39 is irrational and 6.2449979984 is a rounded value.

Where √39 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √39 lies between 6 and 7. 39 is 3 above 36 and 10 below 49, so the root is closer to 6.

• Straight line between 36 and 49: 6.2308 (0.23% low)
• Tangent from 6, i.e. 6 + 3 ÷ 12: 6.2500 (0.08% high)
• Tangent from 7, i.e. 7 − 10 ÷ 14: 6.2857 (0.65% high)

For √39 the tangent at 6 wins, missing by only 0.005. Tangent estimates shine when the number sits close to a perfect square — here 39 is just 3 above 36.

Finding √39 with the Babylonian method

If a guess is too big, 39 ÷ 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√39) in one step.

Start from the nearest whole number, 6 (6² = 36):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √39 = 6.2449979984 to every decimal shown.

√39 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √39 the pattern is [6; 4, 12] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √39 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 − 39y² = 1. Its smallest solution in positive whole numbers is x = 25, y = 4.

√39 in geometry and everyday measurements

• A square room or garden bed covering 39 square feet measures about 6.24 ft (6 ft 3 in) along each wall.
• 39 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 √39 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √39 as its space diagonal.

Square roots near √39 and related roots

• The cube root of 39 is about 3.391211.
• Four times the radicand doubles the root: √156 = 2 × √39 ≈ 12.489996.

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