Square Root of 571

√571 at a glance

Exact value

√571

Decimal (10 places)

23.8956062907

Rounded

23.9 · 23.90 · 23.896

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.895606

Prime factorization

571

Cube root

8.296190

How to simplify √571

571 is a prime number, so its only factors are 1 and 571. There is no perfect-square factor to pull out, which means √571 is already in its simplest radical form.

The square root of any prime is irrational. If √571 were a fraction a/b in lowest terms, then a² = 571b², so 571 would divide a — and then 571 would divide b too, contradicting “lowest terms.” That is why the decimal 23.8956062907 is only a rounded value.

Where √571 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √571 lies between 23 and 24. 571 is 42 above 529 and 5 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.8936 (0.01% low)
• Tangent from 23, i.e. 23 + 42 ÷ 46: 23.9130 (0.07% high)
• Tangent from 24, i.e. 24 − 5 ÷ 48: 23.8958 (0% high)

For √571 the tangent at 24 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 571 is just 5 below 576.

Finding √571 with the Babylonian method

If a guess is too big, 571 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√571) in one step.

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 √571 = 23.8956062907 to every decimal shown.

√571 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √571 the pattern is [23; 1, 8, 1, 1, 2, 1, 1, 1, 15, 3, 2, 1, …] with the block of 42 terms after the semicolon repeating forever (only the first 12 of the 42 are shown). A pattern that never ends is one more proof that √571 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 − 571y² = 1. Its smallest solution in positive whole numbers is x = 181,124,355,061,630,786,130, y = 7,579,818,350,628,982,587 — 21 digits for x, even though 571 is small, which is what makes Pell’s equation famous.

√571 in geometry and everyday measurements

• A square garage floor of 571 square feet measures about 23.9 ft (23 ft 11 in) per side, and its corner-to-corner diagonal is √1142 ≈ 33.8 ft.
• 571 is not a sum of two whole-number squares — 571 is itself a prime that is one less than a multiple of 4, which rules that out — so √571 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 11 × 21 box, because 3² + 11² + 21² = 571.

Square roots near √571 and related roots

• The cube root of 571 is about 8.296190.
• Squaring undoes the root: (√571)² = 571, while 571² = 326,041 — the number whose square root is 571.

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