Square Root of 499

√499 at a glance

Exact value

√499

Decimal (10 places)

22.3383079037

Rounded

22.3 · 22.34 · 22.338

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.338308

Prime factorization

499

Cube root

7.931710

How to simplify √499

499 is a prime number, so its only factors are 1 and 499. There is no perfect-square factor to pull out, which means √499 is already in its simplest radical form.

The square root of any prime is irrational. If √499 were a fraction a/b in lowest terms, then a² = 499b², so 499 would divide a — and then 499 would divide b too, contradicting “lowest terms.” That is why the decimal 22.3383079037 is only a rounded value.

Where √499 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √499 lies between 22 and 23. 499 is 15 above 484 and 30 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.3333 (0.02% low)
• Tangent from 22, i.e. 22 + 15 ÷ 44: 22.3409 (0.01% high)
• Tangent from 23, i.e. 23 − 30 ÷ 46: 22.3478 (0.04% high)

For √499 the tangent at 22 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 499 is just 15 above 484.

Finding √499 with the Babylonian method

If a guess is too big, 499 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√499) in one step.

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 √499 = 22.3383079037 to every decimal shown.

√499 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √499 the pattern is [22; 2, 1, 21, 1, 2, 44] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √499 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 − 499y² = 1. Its smallest solution in positive whole numbers is x = 4,490, y = 201.

√499 in geometry and everyday measurements

• A square garage floor of 499 square feet measures about 22.34 ft (22 ft 4 in) per side, and its corner-to-corner diagonal is √998 ≈ 31.6 ft.
• 499 is not a sum of two whole-number squares — 499 is itself a prime that is one less than a multiple of 4, which rules that out — so √499 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 7 × 21 box, because 3² + 7² + 21² = 499.

Square roots near √499 and related roots

• The cube root of 499 is about 7.931710.
• Squaring undoes the root: (√499)² = 499, while 499² = 249,001 — the number whose square root is 499.

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