Square Root of 65

√65 at a glance

Exact value

√65

Decimal (10 places)

8.0622577483

Rounded

8.1 · 8.06 · 8.062

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.062258

Prime factorization

5 × 13

Cube root

4.020726

How to simplify √65

The prime factorization of 65 is 5 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √65 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 65, 5 and 13 appear an odd number of times, so √65 is irrational and 8.0622577483 is a rounded value.

Where √65 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √65 lies between 8 and 9. 65 is 1 above 64 and 16 below 81, so the root is closer to 8.

• Straight line between 64 and 81: 8.0588 (0.04% low)
• Tangent from 8, i.e. 8 + 1 ÷ 16: 8.0625 (0% high)
• Tangent from 9, i.e. 9 − 16 ÷ 18: 8.1111 (0.61% high)

For √65 the tangent at 8 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 65 is just 1 above 64.

Finding √65 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 65: following the tangent line down to zero simplifies to averaging x with 65 ÷ x.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √65 = 8.0622577483 to every decimal shown.

√65 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √65 the pattern is [8; 16] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 65 is one more than a perfect square (8² + 1). A pattern that never ends is one more proof that √65 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 − 65y² = 1. Its smallest solution in positive whole numbers is x = 129, y = 16. Because the period is odd, the equation with −1 on the right also has a solution: 8² − 65 × 1² = −1.

√65 in geometry and everyday measurements

• A square room or garden bed covering 65 square feet measures about 8.06 ft (8 ft 1 in) along each wall.
• 65 = 1² + 8² = 4² + 7², so by the Pythagorean theorem √65 is the diagonal of rectangles measuring 1 × 8 and 4 × 7 — and the distance between the points (0, 0) and (1, 8) on a grid.

Square roots near √65 and related roots

• The cube root of 65 is about 4.020726.
• Four times the radicand doubles the root: √260 = 2 × √65 ≈ 16.124515.

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