Square Root of 564

√564 at a glance

Exact value

2√141

Decimal (10 places)

23.7486841741

Rounded

23.7 · 23.75 · 23.749

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.748684

Prime factorization

2² × 3 × 47

Cube root

8.262149

How to simplify √564

Look for the largest perfect square that divides 564. Here it is 4 (2²), because 564 = 4 × 141 and 141 has no square factor left:

The prime factorization tells the same story: 564 = 2² × 3 × 47. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 47 stays inside.

Check: (2√141)² = 2² × 141 = 4 × 141 = 564. As a decimal, 2√141 = 2 × 11.874342087 ≈ 23.7486841741.

Where √564 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √564 lies between 23 and 24. 564 is 35 above 529 and 12 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.7447 (0.02% low)
• Tangent from 23, i.e. 23 + 35 ÷ 46: 23.7609 (0.05% high)
• Tangent from 24, i.e. 24 − 12 ÷ 48: 23.7500 (0.01% high)

For √564 the tangent at 24 wins, missing by only 0.0013. Tangent estimates shine when the number sits close to a perfect square — here 564 is just 12 below 576.

Finding √564 with the Babylonian method

Picture a rectangle with an area of 564 and one side x; the other side must be 564 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √564.

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 √564 = 23.7486841741 to every decimal shown.

√564 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √564 the pattern is [23; 1, 2, 1, 46] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √564 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 − 564y² = 1. Its smallest solution in positive whole numbers is x = 95, y = 4.

√564 in geometry and everyday measurements

• A square garage floor of 564 square feet measures about 23.75 ft (23 ft 9 in) per side, and its corner-to-corner diagonal is √1128 ≈ 33.6 ft.
• 564 is not a sum of two whole-number squares — the prime factor 3 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √564 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 22 box, because 4² + 8² + 22² = 564.
• Since √564 = 2√141, a length of √564 is exactly 2 copies of the length √141 laid end to end.

Square roots near √564 and related roots

• The cube root of 564 is about 8.262149.
• Because 564 = 4 × 141, the root is twice √141: 2 × 11.874342 ≈ 23.748684.

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