Square Root of 512

√512 at a glance

Exact value

16√2

Decimal (10 places)

22.6274169980

Rounded

22.6 · 22.63 · 22.627

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.627417

Prime factorization

2⁹

Cube root

8

How to simplify √512

Look for the largest perfect square that divides 512. Here it is 256 (16²), because 512 = 256 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 512 = 2⁹. Each pair of equal primes leaves the radical as one factor, so 2⁴ comes out and 2 stays inside.

512 has 4 square factors (4, 16, 64 and 256). Starting with a smaller one still works but takes more rounds: √512 = 2√128, and √128 can be simplified again. Using 256 straight away finishes in one step.

Check: (16√2)² = 16² × 2 = 256 × 2 = 512. As a decimal, 16√2 = 16 × 1.4142135624 ≈ 22.6274169980.

Where √512 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √512 lies between 22 and 23. 512 is 28 above 484 and 17 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.6222 (0.02% low)
• Tangent from 22, i.e. 22 + 28 ÷ 44: 22.6364 (0.04% high)
• Tangent from 23, i.e. 23 − 17 ÷ 46: 22.6304 (0.01% high)

For √512 the tangent at 23 wins, missing by only 0.003. Tangent estimates shine when the number sits close to a perfect square — here 512 is just 17 below 529.

Finding √512 with the Babylonian method

Picture a rectangle with an area of 512 and one side x; the other side must be 512 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √512.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √512 = 22.6274169980 to every decimal shown.

√512 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √512 the pattern is [22; 1, 1, 1, 2, 6, 11, 6, 2, 1, 1, 1, 44] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √512 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 − 512y² = 1. Its smallest solution in positive whole numbers is x = 665,857, y = 29,427.

√512 in geometry and everyday measurements

• A square garage floor of 512 square feet measures about 22.63 ft (22 ft 8 in) per side, and its corner-to-corner diagonal is √1024 ≈ 32 ft.
• 512 = 16² + 16², so by the Pythagorean theorem √512 is the diagonal of a 16 × 16 rectangle — and the distance between the points (0, 0) and (16, 16) on a grid.
• Since √512 = 16√2, a length of √512 is exactly 16 copies of the length √2 laid end to end.

Square roots near √512 and related roots

• The cube root of 512 is exactly 8 — 512 is a perfect cube as well (8³).
• Because 512 = 4 × 128, the root is twice √128: 2 × 11.313708 ≈ 22.627417.

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