Square Root of 565

√565 at a glance

Exact value

√565

Decimal (10 places)

23.7697286480

Rounded

23.8 · 23.77 · 23.770

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.769729

Prime factorization

5 × 113

Cube root

8.267029

How to simplify √565

The prime factorization of 565 is 5 × 113. Every prime appears only once, so there is no pair to bring outside the radical — √565 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 565, 5 and 113 appear an odd number of times, so √565 is irrational and 23.7697286480 is a rounded value.

Where √565 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √565 lies between 23 and 24. 565 is 36 above 529 and 11 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.7660 (0.02% low)
• Tangent from 23, i.e. 23 + 36 ÷ 46: 23.7826 (0.05% high)
• Tangent from 24, i.e. 24 − 11 ÷ 48: 23.7708 (0% high)

For √565 the tangent at 24 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 565 is just 11 below 576.

Finding √565 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 565: following the tangent line down to zero simplifies to averaging x with 565 ÷ x.

Start from the nearest whole number, 24 (24² = 576):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √565 = 23.7697286480 to every decimal shown.

√565 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √565 the pattern is [23; 1, 3, 2, 1, 11, 5, 5, 11, 1, 2, 3, 1, …] with the block of 13 terms after the semicolon repeating forever (only the first 12 of the 13 are shown). A pattern that never ends is one more proof that √565 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 − 565y² = 1. Its smallest solution in positive whole numbers is x = 435,259,412,378,569, y = 18,311,501,103,948 — 15 digits for x, even though 565 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 14,752,278² − 565 × 620,633² = −1.

√565 in geometry and everyday measurements

• A square garage floor of 565 square feet measures about 23.77 ft (23 ft 9 in) per side, and its corner-to-corner diagonal is √1130 ≈ 33.6 ft.
• 565 = 6² + 23² = 9² + 22², so by the Pythagorean theorem √565 is the diagonal of rectangles measuring 6 × 23 and 9 × 22 — and the distance between the points (0, 0) and (6, 23) on a grid.

Square roots near √565 and related roots

• The cube root of 565 is about 8.267029.
• Squaring undoes the root: (√565)² = 565, while 565² = 319,225 — the number whose square root is 565.

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