Square Root of 549

√549 at a glance

Exact value

3√61

Decimal (10 places)

23.4307490277

Rounded

23.4 · 23.43 · 23.431

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.430749

Prime factorization

3² × 61

Cube root

8.188244

How to simplify √549

Look for the largest perfect square that divides 549. Here it is 9 (3²), because 549 = 9 × 61 and 61 has no square factor left:

The prime factorization tells the same story: 549 = 3² × 61. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 61 stays inside.

Check: (3√61)² = 3² × 61 = 9 × 61 = 549. As a decimal, 3√61 = 3 × 7.8102496759 ≈ 23.4307490277.

Where √549 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √549 lies between 23 and 24. 549 is 20 above 529 and 27 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.4255 (0.02% low)
• Tangent from 23, i.e. 23 + 20 ÷ 46: 23.4348 (0.02% high)
• Tangent from 24, i.e. 24 − 27 ÷ 48: 23.4375 (0.03% high)

For √549 the tangent at 23 wins, missing by only 0.004. Tangent estimates shine when the number sits close to a perfect square — here 549 is just 20 above 529.

Finding √549 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 549: following the tangent line down to zero simplifies to averaging x with 549 ÷ x.

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 √549 = 23.4307490277 to every decimal shown.

√549 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √549 the pattern is [23; 2, 3, 9, 11, 1, 1, 1, 1, 4, 1, 1, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √549 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 − 549y² = 1. Its smallest solution in positive whole numbers is x = 1,766,319,049, y = 75,384,660.

√549 in geometry and everyday measurements

• A square garage floor of 549 square feet measures about 23.43 ft (23 ft 5 in) per side, and its corner-to-corner diagonal is √1098 ≈ 33.1 ft.
• 549 = 15² + 18², so by the Pythagorean theorem √549 is the diagonal of a 15 × 18 rectangle — and the distance between the points (0, 0) and (15, 18) on a grid.
• Since √549 = 3√61, a length of √549 is exactly 3 copies of the length √61 laid end to end.

Square roots near √549 and related roots

• The cube root of 549 is about 8.188244.
• Squaring undoes the root: (√549)² = 549, while 549² = 301,401 — the number whose square root is 549.

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