Square Root of 572

√572 at a glance

Exact value

2√143

Decimal (10 places)

23.9165214862

Rounded

23.9 · 23.92 · 23.917

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.916521

Prime factorization

2² × 11 × 13

Cube root

8.301031

How to simplify √572

Look for the largest perfect square that divides 572. Here it is 4 (2²), because 572 = 4 × 143 and 143 has no square factor left:

The prime factorization tells the same story: 572 = 2² × 11 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 11 × 13 stays inside.

Check: (2√143)² = 2² × 143 = 4 × 143 = 572. As a decimal, 2√143 = 2 × 11.9582607431 ≈ 23.9165214862.

Where √572 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √572 lies between 23 and 24. 572 is 43 above 529 and 4 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.9149 (0.01% low)
• Tangent from 23, i.e. 23 + 43 ÷ 46: 23.9348 (0.08% high)
• Tangent from 24, i.e. 24 − 4 ÷ 48: 23.9167 (0% high)

For √572 the tangent at 24 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 572 is just 4 below 576.

Finding √572 with the Babylonian method

Picture a rectangle with an area of 572 and one side x; the other side must be 572 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √572.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √572 = 23.9165214862 to every decimal shown.

√572 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √572 the pattern is [23; 1, 10, 1, 46] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √572 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 − 572y² = 1. Its smallest solution in positive whole numbers is x = 287, y = 12.

√572 in geometry and everyday measurements

• A square garage floor of 572 square feet measures about 23.92 ft (23 ft 11 in) per side, and its corner-to-corner diagonal is √1144 ≈ 33.8 ft.
• 572 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √572 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √572 as its space diagonal.
• Since √572 = 2√143, a length of √572 is exactly 2 copies of the length √143 laid end to end.

Square roots near √572 and related roots

• The cube root of 572 is about 8.301031.
• Because 572 = 4 × 143, the root is twice √143: 2 × 11.958261 ≈ 23.916521.

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