Square Root of 513

√513 at a glance

Exact value

3√57

Decimal (10 places)

22.6495033058

Rounded

22.6 · 22.65 · 22.650

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.649503

Prime factorization

3³ × 19

Cube root

8.005205

How to simplify √513

Look for the largest perfect square that divides 513. Here it is 9 (3²), because 513 = 9 × 57 and 57 has no square factor left:

The prime factorization tells the same story: 513 = 3³ × 19. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 19 stays inside.

Check: (3√57)² = 3² × 57 = 9 × 57 = 513. As a decimal, 3√57 = 3 × 7.5498344353 ≈ 22.6495033058.

Where √513 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √513 lies between 22 and 23. 513 is 29 above 484 and 16 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.6444 (0.02% low)
• Tangent from 22, i.e. 22 + 29 ÷ 44: 22.6591 (0.04% high)
• Tangent from 23, i.e. 23 − 16 ÷ 46: 22.6522 (0.01% high)

For √513 the tangent at 23 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 513 is just 16 below 529.

Finding √513 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 513: following the tangent line down to zero simplifies to averaging x with 513 ÷ x.

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 √513 = 22.6495033058 to every decimal shown.

√513 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √513 the pattern is [22; 1, 1, 1, 5, 1, 4, 5, 2, 5, 4, 1, 5, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √513 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 − 513y² = 1. Its smallest solution in positive whole numbers is x = 13,771,351, y = 608,020.

√513 in geometry and everyday measurements

• A square garage floor of 513 square feet measures about 22.65 ft (22 ft 8 in) per side, and its corner-to-corner diagonal is √1026 ≈ 32 ft.
• 513 is not a sum of two whole-number squares — the prime factor 3 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √513 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 16 × 16 box, because 1² + 16² + 16² = 513.
• Since √513 = 3√57, a length of √513 is exactly 3 copies of the length √57 laid end to end.

Square roots near √513 and related roots

• The cube root of 513 is about 8.005205.
• Squaring undoes the root: (√513)² = 513, while 513² = 263,169 — the number whose square root is 513.

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