Square Root of 524

√524 at a glance

Exact value

2√131

Decimal (10 places)

22.8910462845

Rounded

22.9 · 22.89 · 22.891

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.891046

Prime factorization

2² × 131

Cube root

8.062018

How to simplify √524

Look for the largest perfect square that divides 524. Here it is 4 (2²), because 524 = 4 × 131 and 131 has no square factor left:

The prime factorization tells the same story: 524 = 2² × 131. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 131 stays inside.

Check: (2√131)² = 2² × 131 = 4 × 131 = 524. As a decimal, 2√131 = 2 × 11.4455231423 ≈ 22.8910462845.

Where √524 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √524 lies between 22 and 23. 524 is 40 above 484 and 5 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.8889 (0.01% low)
• Tangent from 22, i.e. 22 + 40 ÷ 44: 22.9091 (0.08% high)
• Tangent from 23, i.e. 23 − 5 ÷ 46: 22.8913 (0% high)

For √524 the tangent at 23 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 524 is just 5 below 529.

Finding √524 with the Babylonian method

Picture a rectangle with an area of 524 and one side x; the other side must be 524 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √524.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √524 = 22.8910462845 to every decimal shown.

√524 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √524 the pattern is [22; 1, 8, 5, 1, 1, 1, 1, 2, 1, 10, 1, 2, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √524 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 − 524y² = 1. Its smallest solution in positive whole numbers is x = 225,144,199, y = 9,835,470.

√524 in geometry and everyday measurements

• A square garage floor of 524 square feet measures about 22.89 ft (22 ft 11 in) per side, and its corner-to-corner diagonal is √1048 ≈ 32.4 ft.
• 524 is not a sum of two whole-number squares — the prime factor 131 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √524 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 22 box, because 2² + 6² + 22² = 524.
• Since √524 = 2√131, a length of √524 is exactly 2 copies of the length √131 laid end to end.

Square roots near √524 and related roots

• The cube root of 524 is about 8.062018.
• Because 524 = 4 × 131, the root is twice √131: 2 × 11.445523 ≈ 22.891046.

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