Square Root of 552

√552 at a glance

Exact value

2√138

Decimal (10 places)

23.4946802489

Rounded

23.5 · 23.49 · 23.495

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.494680

Prime factorization

2³ × 3 × 23

Cube root

8.203132

How to simplify √552

Look for the largest perfect square that divides 552. Here it is 4 (2²), because 552 = 4 × 138 and 138 has no square factor left:

The prime factorization tells the same story: 552 = 2³ × 3 × 23. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 23 stays inside.

Check: (2√138)² = 2² × 138 = 4 × 138 = 552. As a decimal, 2√138 = 2 × 11.7473401245 ≈ 23.4946802489.

Where √552 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √552 lies between 23 and 24. 552 is 23 above 529 and 24 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.4894 (0.02% low)
• Tangent from 23, i.e. 23 + 23 ÷ 46: 23.5000 (0.02% high)
• Tangent from 24, i.e. 24 − 24 ÷ 48: 23.5000 (0.02% high)

For √552 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √552 with the Babylonian method

Picture a rectangle with an area of 552 and one side x; the other side must be 552 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √552.

Start from the nearest whole number, 23 (23² = 529):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √552 = 23.4946802489 to every decimal shown.

√552 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √552 the pattern is [23; 2, 46] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √552 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 − 552y² = 1. Its smallest solution in positive whole numbers is x = 47, y = 2.

√552 in geometry and everyday measurements

• A square garage floor of 552 square feet measures about 23.49 ft (23 ft 6 in) per side, and its corner-to-corner diagonal is √1104 ≈ 33.2 ft.
• 552 is not a sum of two whole-number squares — the prime factor 3 and 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √552 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 22 box, because 2² + 8² + 22² = 552.
• Since √552 = 2√138, a length of √552 is exactly 2 copies of the length √138 laid end to end.

Square roots near √552 and related roots

• The cube root of 552 is about 8.203132.
• Because 552 = 4 × 138, the root is twice √138: 2 × 11.74734 ≈ 23.49468.

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