Square Root of 508

√508 at a glance

Exact value

2√127

Decimal (10 places)

22.5388553392

Rounded

22.5 · 22.54 · 22.539

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.538855

Prime factorization

2² × 127

Cube root

7.979112

How to simplify √508

Look for the largest perfect square that divides 508. Here it is 4 (2²), because 508 = 4 × 127 and 127 has no square factor left:

The prime factorization tells the same story: 508 = 2² × 127. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 127 stays inside.

Check: (2√127)² = 2² × 127 = 4 × 127 = 508. As a decimal, 2√127 = 2 × 11.2694276696 ≈ 22.5388553392.

Where √508 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √508 lies between 22 and 23. 508 is 24 above 484 and 21 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.5333 (0.02% low)
• Tangent from 22, i.e. 22 + 24 ÷ 44: 22.5455 (0.03% high)
• Tangent from 23, i.e. 23 − 21 ÷ 46: 22.5435 (0.02% high)

For √508 the tangent at 23 wins, missing by only 0.0046. Tangent estimates shine when the number sits close to a perfect square — here 508 is just 21 below 529.

Finding √508 with the Babylonian method

Picture a rectangle with an area of 508 and one side x; the other side must be 508 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √508.

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 √508 = 22.5388553392 to every decimal shown.

√508 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √508 the pattern is [22; 1, 1, 5, 1, 14, 5, 1, 1, 3, 4, 1, 2, …] with the block of 32 terms after the semicolon repeating forever (only the first 12 of the 32 are shown). A pattern that never ends is one more proof that √508 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 − 508y² = 1. Its smallest solution in positive whole numbers is x = 44,757,606,858,751, y = 1,985,797,689,600 — 14 digits for x, even though 508 is small, which is what makes Pell’s equation famous.

√508 in geometry and everyday measurements

• A square garage floor of 508 square feet measures about 22.54 ft (22 ft 6 in) per side, and its corner-to-corner diagonal is √1016 ≈ 31.9 ft.
• 508 is not a sum of two whole-number squares — the prime factor 127 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √508 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √508 as its space diagonal.
• Since √508 = 2√127, a length of √508 is exactly 2 copies of the length √127 laid end to end.

Square roots near √508 and related roots

• The cube root of 508 is about 7.979112.
• Because 508 = 4 × 127, the root is twice √127: 2 × 11.269428 ≈ 22.538855.

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