Square Root of 556

√556 at a glance

Exact value

2√139

Decimal (10 places)

23.5796522451

Rounded

23.6 · 23.58 · 23.580

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.579652

Prime factorization

2² × 139

Cube root

8.222899

How to simplify √556

Look for the largest perfect square that divides 556. Here it is 4 (2²), because 556 = 4 × 139 and 139 has no square factor left:

The prime factorization tells the same story: 556 = 2² × 139. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 139 stays inside.

Check: (2√139)² = 2² × 139 = 4 × 139 = 556. As a decimal, 2√139 = 2 × 11.7898261226 ≈ 23.5796522451.

Where √556 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √556 lies between 23 and 24. 556 is 27 above 529 and 20 below 576, so the root is closer to 24.

• Straight line between 529 and 576: 23.5745 (0.02% low)
• Tangent from 23, i.e. 23 + 27 ÷ 46: 23.5870 (0.03% high)
• Tangent from 24, i.e. 24 − 20 ÷ 48: 23.5833 (0.02% high)

For √556 the tangent at 24 wins, missing by only 0.0037. Tangent estimates shine when the number sits close to a perfect square — here 556 is just 20 below 576.

Finding √556 with the Babylonian method

Picture a rectangle with an area of 556 and one side x; the other side must be 556 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √556.

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 √556 = 23.5796522451 to every decimal shown.

√556 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √556 the pattern is [23; 1, 1, 2, 1, 1, 1, 3, 3, 2, 1, 5, 5, …] with the block of 36 terms after the semicolon repeating forever (only the first 12 of the 36 are shown). A pattern that never ends is one more proof that √556 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 − 556y² = 1. Its smallest solution in positive whole numbers is x = 12,032,115,501,124,999, y = 510,275,358,434,250 — 17 digits for x, even though 556 is small, which is what makes Pell’s equation famous.

√556 in geometry and everyday measurements

• A square garage floor of 556 square feet measures about 23.58 ft (23 ft 7 in) per side, and its corner-to-corner diagonal is √1112 ≈ 33.3 ft.
• 556 is not a sum of two whole-number squares — the prime factor 139 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √556 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 6 × 22 box, because 6² + 6² + 22² = 556.
• Since √556 = 2√139, a length of √556 is exactly 2 copies of the length √139 laid end to end.

Square roots near √556 and related roots

• The cube root of 556 is about 8.222899.
• Because 556 = 4 × 139, the root is twice √139: 2 × 11.789826 ≈ 23.579652.

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