Square Root of 521

√521 at a glance

Exact value

√521

Decimal (10 places)

22.8254244210

Rounded

22.8 · 22.83 · 22.825

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.825424

Prime factorization

521

Cube root

8.046603

How to simplify √521

521 is a prime number, so its only factors are 1 and 521. There is no perfect-square factor to pull out, which means √521 is already in its simplest radical form.

The square root of any prime is irrational. If √521 were a fraction a/b in lowest terms, then a² = 521b², so 521 would divide a — and then 521 would divide b too, contradicting “lowest terms.” That is why the decimal 22.8254244210 is only a rounded value.

Where √521 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √521 lies between 22 and 23. 521 is 37 above 484 and 8 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.8222 (0.01% low)
• Tangent from 22, i.e. 22 + 37 ÷ 44: 22.8409 (0.07% high)
• Tangent from 23, i.e. 23 − 8 ÷ 46: 22.8261 (0% high)

For √521 the tangent at 23 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 521 is just 8 below 529.

Finding √521 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 521: following the tangent line down to zero simplifies to averaging x with 521 ÷ x.

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 √521 = 22.8254244210 to every decimal shown.

√521 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √521 the pattern is [22; 1, 4, 1, 2, 1, 2, 8, 1, 3, 3, 1, 8, …] with the block of 19 terms after the semicolon repeating forever (only the first 12 of the 19 are shown). A pattern that never ends is one more proof that √521 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 − 521y² = 1. Its smallest solution in positive whole numbers is x = 32,961,431,500,035,201, y = 1,444,066,532,654,320 — 17 digits for x, even though 521 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 128,377,240² − 521 × 5,624,309² = −1.

√521 in geometry and everyday measurements

• A square garage floor of 521 square feet measures about 22.83 ft (22 ft 10 in) per side, and its corner-to-corner diagonal is √1042 ≈ 32.3 ft.
• 521 = 11² + 20², so by the Pythagorean theorem √521 is the diagonal of a 11 × 20 rectangle — and the distance between the points (0, 0) and (11, 20) on a grid.

Square roots near √521 and related roots

• The cube root of 521 is about 8.046603.
• Squaring undoes the root: (√521)² = 521, while 521² = 271,441 — the number whose square root is 521.

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