Square Root of 517

√517 at a glance

Exact value

√517

Decimal (10 places)

22.7376340018

Rounded

22.7 · 22.74 · 22.738

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.737634

Prime factorization

11 × 47

Cube root

8.025957

How to simplify √517

The prime factorization of 517 is 11 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √517 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 517, 11 and 47 appear an odd number of times, so √517 is irrational and 22.7376340018 is a rounded value.

Where √517 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √517 lies between 22 and 23. 517 is 33 above 484 and 12 below 529, so the root is closer to 23.

• Straight line between 484 and 529: 22.7333 (0.02% low)
• Tangent from 22, i.e. 22 + 33 ÷ 44: 22.7500 (0.05% high)
• Tangent from 23, i.e. 23 − 12 ÷ 46: 22.7391 (0.01% high)

For √517 the tangent at 23 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 517 is just 12 below 529.

Finding √517 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 517: following the tangent line down to zero simplifies to averaging x with 517 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √517 = 22.7376340018 to every decimal shown.

√517 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √517 the pattern is [22; 1, 2, 1, 4, 3, 3, 2, 10, 1, 14, 4, 14, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √517 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 − 517y² = 1. Its smallest solution in positive whole numbers is x = 590,968,985,399, y = 25,990,786,260.

√517 in geometry and everyday measurements

• A square garage floor of 517 square feet measures about 22.74 ft (22 ft 9 in) per side, and its corner-to-corner diagonal is √1034 ≈ 32.2 ft.
• 517 is not a sum of two whole-number squares — the prime factor 11 and 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √517 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 9 × 20 box, because 6² + 9² + 20² = 517.

Square roots near √517 and related roots

• The cube root of 517 is about 8.025957.
• Squaring undoes the root: (√517)² = 517, while 517² = 267,289 — the number whose square root is 517.

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