Square Root of 531

√531 at a glance

Exact value

3√59

Decimal (10 places)

23.0434372436

Rounded

23.0 · 23.04 · 23.043

Perfect square?

No — between 23² and 24²

Rational?

Irrational

Both square roots

±23.043437

Prime factorization

3² × 59

Cube root

8.097759

How to simplify √531

Look for the largest perfect square that divides 531. Here it is 9 (3²), because 531 = 9 × 59 and 59 has no square factor left:

The prime factorization tells the same story: 531 = 3² × 59. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 59 stays inside.

Check: (3√59)² = 3² × 59 = 9 × 59 = 531. As a decimal, 3√59 = 3 × 7.6811457479 ≈ 23.0434372436.

Where √531 sits between perfect squares

529 = 23² and 576 = 24² are the nearest perfect squares, so √531 lies between 23 and 24. 531 is 2 above 529 and 45 below 576, so the root is closer to 23.

• Straight line between 529 and 576: 23.0426 (0% low)
• Tangent from 23, i.e. 23 + 2 ÷ 46: 23.0435 (0% high)
• Tangent from 24, i.e. 24 − 45 ÷ 48: 23.0625 (0.08% high)

For √531 the tangent at 23 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 531 is just 2 above 529.

Finding √531 with the Babylonian method

If a guess is too big, 531 ÷ 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√531) in one step.

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

Because the starting guess was already close, two steps are enough to match √531 = 23.0434372436 to every decimal shown.

√531 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √531 the pattern is [23; 23, 46] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √531 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 − 531y² = 1. Its smallest solution in positive whole numbers is x = 530, y = 23.

√531 in geometry and everyday measurements

• A square garage floor of 531 square feet measures about 23.04 ft (23 ft 1 in) per side, and its corner-to-corner diagonal is √1062 ≈ 32.6 ft.
• 531 is not a sum of two whole-number squares — the prime factor 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √531 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 23 box, because 1² + 1² + 23² = 531.
• Since √531 = 3√59, a length of √531 is exactly 3 copies of the length √59 laid end to end.

Square roots near √531 and related roots

• The cube root of 531 is about 8.097759.
• Squaring undoes the root: (√531)² = 531, while 531² = 281,961 — the number whose square root is 531.

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