Square Root of 568

√568 at a glance

Exact value

2√142

Decimal (10 places)

23.8327505756

Rounded

23.8 · 23.83 · 23.833

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.832751

Prime factorization

2³ × 71

Cube root

8.281635

How to simplify √568

Look for the largest perfect square that divides 568. Here it is 4 (2²), because 568 = 4 × 142 and 142 has no square factor left:

The prime factorization tells the same story: 568 = 2³ × 71. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 71 stays inside.

Check: (2√142)² = 2² × 142 = 4 × 142 = 568. As a decimal, 2√142 = 2 × 11.9163752878 ≈ 23.8327505756.

Where √568 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √568 lies between 23 and 24. 568 is 39 above 529 and 8 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.8298 (0.01% low)
• Tangent from 23, i.e. 23 + 39 ÷ 46: 23.8478 (0.06% high)
• Tangent from 24, i.e. 24 − 8 ÷ 48: 23.8333 (0% high)

For √568 the tangent at 24 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 568 is just 8 below 576.

Finding √568 with the Babylonian method

Picture a rectangle with an area of 568 and one side x; the other side must be 568 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √568.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √568 = 23.8327505756 to every decimal shown.

√568 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √568 the pattern is [23; 1, 4, 1, 46] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √568 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 − 568y² = 1. Its smallest solution in positive whole numbers is x = 143, y = 6.

√568 in geometry and everyday measurements

• A square garage floor of 568 square feet measures about 23.83 ft (23 ft 10 in) per side, and its corner-to-corner diagonal is √1136 ≈ 33.7 ft.
• 568 is not a sum of two whole-number squares — the prime factor 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √568 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 10 × 12 × 18 box, because 10² + 12² + 18² = 568.
• Since √568 = 2√142, a length of √568 is exactly 2 copies of the length √142 laid end to end.

Square roots near √568 and related roots

• The cube root of 568 is about 8.281635.
• Because 568 = 4 × 142, the root is twice √142: 2 × 11.916375 ≈ 23.832751.

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