Square Root of 555

√555 at a glance

Exact value

√555

Decimal (10 places)

23.5584379788

Rounded

23.6 · 23.56 · 23.558

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.558438

Prime factorization

3 × 5 × 37

Cube root

8.217966

How to simplify √555

The prime factorization of 555 is 3 × 5 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √555 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 555, 3, 5 and 37 appear an odd number of times, so √555 is irrational and 23.5584379788 is a rounded value.

Where √555 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √555 lies between 23 and 24. 555 is 26 above 529 and 21 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.5532 (0.02% low)
• Tangent from 23, i.e. 23 + 26 ÷ 46: 23.5652 (0.03% high)
• Tangent from 24, i.e. 24 − 21 ÷ 48: 23.5625 (0.02% high)

For √555 the tangent at 24 wins, missing by only 0.0041. Tangent estimates shine when the number sits close to a perfect square — here 555 is just 21 below 576.

Finding √555 with the Babylonian method

If a guess is too big, 555 ÷ 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√555) in one step.

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 √555 = 23.5584379788 to every decimal shown.

√555 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √555 the pattern is [23; 1, 1, 3, 1, 3, 1, 1, 46] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √555 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 − 555y² = 1. Its smallest solution in positive whole numbers is x = 1,814, y = 77.

√555 in geometry and everyday measurements

• A square garage floor of 555 square feet measures about 23.56 ft (23 ft 7 in) per side, and its corner-to-corner diagonal is √1110 ≈ 33.3 ft.
• 555 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √555 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 5 × 23 box, because 1² + 5² + 23² = 555.

Square roots near √555 and related roots

• The cube root of 555 is about 8.217966.
• Squaring undoes the root: (√555)² = 555, while 555² = 308,025 — the number whose square root is 555.

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