Square Root of 496

√496 at a glance

Exact value

4√31

Decimal (10 places)

22.2710574513

Rounded

22.3 · 22.27 · 22.271

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.271057

Prime factorization

2⁴ × 31

Cube root

7.915783

How to simplify √496

Look for the largest perfect square that divides 496. Here it is 16 (4²), because 496 = 16 × 31 and 31 has no square factor left:

The prime factorization tells the same story: 496 = 2⁴ × 31. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 31 stays inside.

496 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √496 = 2√124, and √124 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√31)² = 4² × 31 = 16 × 31 = 496. As a decimal, 4√31 = 4 × 5.5677643628 ≈ 22.2710574513.

Where √496 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √496 lies between 22 and 23. 496 is 12 above 484 and 33 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.2667 (0.02% low)
• Tangent from 22, i.e. 22 + 12 ÷ 44: 22.2727 (0.01% high)
• Tangent from 23, i.e. 23 − 33 ÷ 46: 22.2826 (0.05% high)

For √496 the tangent at 22 wins, missing by only 0.0017. Tangent estimates shine when the number sits close to a perfect square — here 496 is just 12 above 484.

Finding √496 with the Babylonian method

Picture a rectangle with an area of 496 and one side x; the other side must be 496 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √496.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √496 = 22.2710574513 to every decimal shown.

√496 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √496 the pattern is [22; 3, 1, 2, 4, 1, 1, 2, 2, 2, 1, 1, 4, …] 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 √496 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 − 496y² = 1. Its smallest solution in positive whole numbers is x = 4,620,799, y = 207,480.

√496 in geometry and everyday measurements

• A square garage floor of 496 square feet measures about 22.27 ft (22 ft 3 in) per side, and its corner-to-corner diagonal is √992 ≈ 31.5 ft.
• 496 is not a sum of two whole-number squares — the prime factor 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √496 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √496 as its space diagonal.
• Since √496 = 4√31, a length of √496 is exactly 4 copies of the length √31 laid end to end.

Square roots near √496 and related roots

• The cube root of 496 is about 7.915783.
• Because 496 = 4 × 124, the root is twice √124: 2 × 11.135529 ≈ 22.271057.

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