Square Root of 557

√557 at a glance

Exact value

√557

Decimal (10 places)

23.6008474424

Rounded

23.6 · 23.60 · 23.601

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.600847

Prime factorization

557

Cube root

8.227825

How to simplify √557

557 is a prime number, so its only factors are 1 and 557. There is no perfect-square factor to pull out, which means √557 is already in its simplest radical form.

The square root of any prime is irrational. If √557 were a fraction a/b in lowest terms, then a² = 557b², so 557 would divide a — and then 557 would divide b too, contradicting “lowest terms.” That is why the decimal 23.6008474424 is only a rounded value.

Where √557 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √557 lies between 23 and 24. 557 is 28 above 529 and 19 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.5957 (0.02% low)
• Tangent from 23, i.e. 23 + 28 ÷ 46: 23.6087 (0.03% high)
• Tangent from 24, i.e. 24 − 19 ÷ 48: 23.6042 (0.01% high)

For √557 the tangent at 24 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 557 is just 19 below 576.

Finding √557 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 557: following the tangent line down to zero simplifies to averaging x with 557 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √557 = 23.6008474424 to every decimal shown.

√557 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √557 the pattern is [23; 1, 1, 1, 1, 46] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √557 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 − 557y² = 1. Its smallest solution in positive whole numbers is x = 27,849, y = 1,180. Because the period is odd, the equation with −1 on the right also has a solution: 118² − 557 × 5² = −1.

√557 in geometry and everyday measurements

• A square garage floor of 557 square feet measures about 23.6 ft (23 ft 7 in) per side, and its corner-to-corner diagonal is √1114 ≈ 33.4 ft.
• 557 = 14² + 19², so by the Pythagorean theorem √557 is the diagonal of a 14 × 19 rectangle — and the distance between the points (0, 0) and (14, 19) on a grid.

Square roots near √557 and related roots

• The cube root of 557 is about 8.227825.
• Squaring undoes the root: (√557)² = 557, while 557² = 310,249 — the number whose square root is 557.

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