Square Root of 69

√69 at a glance

Exact value

√69

Decimal (10 places)

8.3066238629

Rounded

8.3 · 8.31 · 8.307

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.306624

Prime factorization

3 × 23

Cube root

4.101566

How to simplify √69

The prime factorization of 69 is 3 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √69 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 69, 3 and 23 appear an odd number of times, so √69 is irrational and 8.3066238629 is a rounded value.

Where √69 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √69 lies between 8 and 9. 69 is 5 above 64 and 12 below 81, so the root is closer to 8.

• Straight line between 64 and 81: 8.2941 (0.15% low)
• Tangent from 8, i.e. 8 + 5 ÷ 16: 8.3125 (0.07% high)
• Tangent from 9, i.e. 9 − 12 ÷ 18: 8.3333 (0.32% high)

For √69 the tangent at 8 wins, missing by only 0.0059. Tangent estimates shine when the number sits close to a perfect square — here 69 is just 5 above 64.

Finding √69 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 69: following the tangent line down to zero simplifies to averaging x with 69 ÷ x.

Start from the nearest whole number, 8 (8² = 64):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √69 = 8.3066238629 to every decimal shown.

√69 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √69 the pattern is [8; 3, 3, 1, 4, 1, 3, 3, 16] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √69 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 − 69y² = 1. Its smallest solution in positive whole numbers is x = 7,775, y = 936.

√69 in geometry and everyday measurements

• A square room or garden bed covering 69 square feet measures about 8.31 ft (8 ft 4 in) along each wall.
• 69 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √69 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 8 box, because 1² + 2² + 8² = 69.

Square roots near √69 and related roots

• The cube root of 69 is about 4.101566.
• Four times the radicand doubles the root: √276 = 2 × √69 ≈ 16.613248.

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