Square Root of 493

√493 at a glance

Exact value

√493

Decimal (10 places)

22.2036033112

Rounded

22.2 · 22.20 · 22.204

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.203603

Prime factorization

17 × 29

Cube root

7.899792

How to simplify √493

The prime factorization of 493 is 17 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √493 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 493, 17 and 29 appear an odd number of times, so √493 is irrational and 22.2036033112 is a rounded value.

Where √493 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √493 lies between 22 and 23. 493 is 9 above 484 and 36 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.2000 (0.02% low)
• Tangent from 22, i.e. 22 + 9 ÷ 44: 22.2045 (0% high)
• Tangent from 23, i.e. 23 − 36 ÷ 46: 22.2174 (0.06% high)

For √493 the tangent at 22 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 493 is just 9 above 484.

Finding √493 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 493: following the tangent line down to zero simplifies to averaging x with 493 ÷ x.

Start from the nearest whole number, 22 (22² = 484):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √493 = 22.2036033112 to every decimal shown.

√493 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √493 the pattern is [22; 4, 1, 10, 3, 3, 10, 1, 4, 44] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √493 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 − 493y² = 1. Its smallest solution in positive whole numbers is x = 935,662,752,649, y = 42,140,131,020. Because the period is odd, the equation with −1 on the right also has a solution: 683,982² − 493 × 30,805² = −1.

√493 in geometry and everyday measurements

• A square garage floor of 493 square feet measures about 22.2 ft (22 ft 2 in) per side, and its corner-to-corner diagonal is √986 ≈ 31.4 ft.
• 493 = 3² + 22² = 13² + 18², so by the Pythagorean theorem √493 is the diagonal of rectangles measuring 3 × 22 and 13 × 18 — and the distance between the points (0, 0) and (3, 22) on a grid.

Square roots near √493 and related roots

• The cube root of 493 is about 7.899792.
• Squaring undoes the root: (√493)² = 493, while 493² = 243,049 — the number whose square root is 493.

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