Square Root of 765

√765 at a glance

Exact value

3√85

Decimal (10 places)

27.6586333719

Rounded

27.7 · 27.66 · 27.659

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.658633

Prime factorization

3² × 5 × 17

Cube root

9.145774

How to simplify √765

Look for the largest perfect square that divides 765. Here it is 9 (3²), because 765 = 9 × 85 and 85 has no square factor left:

The prime factorization tells the same story: 765 = 3² × 5 × 17. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 5 × 17 stays inside.

Check: (3√85)² = 3² × 85 = 9 × 85 = 765. As a decimal, 3√85 = 3 × 9.2195444573 ≈ 27.6586333719.

Where √765 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √765 lies between 27 and 28. 765 is 36 above 729 and 19 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.6545 (0.01% low)
• Tangent from 27, i.e. 27 + 36 ÷ 54: 27.6667 (0.03% high)
• Tangent from 28, i.e. 28 − 19 ÷ 56: 27.6607 (0.01% high)

For √765 the tangent at 28 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 765 is just 19 below 784.

Finding √765 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 765: following the tangent line down to zero simplifies to averaging x with 765 ÷ x.

Start from the nearest whole number, 28 (28² = 784):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √765 = 27.6586333719 to every decimal shown.

√765 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √765 the pattern is [27; 1, 1, 1, 13, 6, 13, 1, 1, 1, 54] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √765 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 − 765y² = 1. Its smallest solution in positive whole numbers is x = 285,769, y = 10,332.

√765 in geometry and everyday measurements

• 765 square feet is 71.1 m². Laid out as a square — a small house footprint or a lot — it is about 27.66 ft (27 ft 8 in) on a side.
• 765 = 6² + 27² = 18² + 21², so by the Pythagorean theorem √765 is the diagonal of rectangles measuring 6 × 27 and 18 × 21 — and the distance between the points (0, 0) and (6, 27) on a grid.
• Since √765 = 3√85, a length of √765 is exactly 3 copies of the length √85 laid end to end.

Square roots near √765 and related roots

• The cube root of 765 is about 9.145774.
• Squaring undoes the root: (√765)² = 765, while 765² = 585,225 — the number whose square root is 765.

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