Square Root of 504

√504 at a glance

Exact value

6√14

Decimal (10 places)

22.4499443206

Rounded

22.4 · 22.45 · 22.450

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.449944

Prime factorization

2³ × 3² × 7

Cube root

7.958114

How to simplify √504

Look for the largest perfect square that divides 504. Here it is 36 (6²), because 504 = 36 × 14 and 14 has no square factor left:

The prime factorization tells the same story: 504 = 2³ × 3² × 7. Each pair of equal primes leaves the radical as one factor, so 2 × 3 comes out and 2 × 7 stays inside.

504 has 3 square factors (4, 9 and 36). Starting with a smaller one still works but takes more rounds: √504 = 2√126, and √126 can be simplified again. Using 36 straight away finishes in one step.

Check: (6√14)² = 6² × 14 = 36 × 14 = 504. As a decimal, 6√14 = 6 × 3.7416573868 ≈ 22.4499443206.

Where √504 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √504 lies between 22 and 23. 504 is 20 above 484 and 25 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.4444 (0.02% low)
• Tangent from 22, i.e. 22 + 20 ÷ 44: 22.4545 (0.02% high)
• Tangent from 23, i.e. 23 − 25 ÷ 46: 22.4565 (0.03% high)

For √504 the tangent at 22 wins, missing by only 0.0046. Tangent estimates shine when the number sits close to a perfect square — here 504 is just 20 above 484.

Finding √504 with the Babylonian method

Picture a rectangle with an area of 504 and one side x; the other side must be 504 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √504.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √504 = 22.4499443206 to every decimal shown.

√504 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √504 the pattern is [22; 2, 4, 2, 44] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √504 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 − 504y² = 1. Its smallest solution in positive whole numbers is x = 449, y = 20.

√504 in geometry and everyday measurements

• A square garage floor of 504 square feet measures about 22.45 ft (22 ft 5 in) per side, and its corner-to-corner diagonal is √1008 ≈ 31.7 ft.
• 504 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √504 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 22 box, because 2² + 4² + 22² = 504.
• Since √504 = 6√14, a length of √504 is exactly 6 copies of the length √14 laid end to end.

Square roots near √504 and related roots

• The cube root of 504 is about 7.958114.
• Because 504 = 4 × 126, the root is twice √126: 2 × 11.224972 ≈ 22.449944.

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