Square Root of 488

√488 at a glance

Exact value

2√122

Decimal (10 places)

22.0907220344

Rounded

22.1 · 22.09 · 22.091

Perfect square?

No — between 22² and 23²

Rational?

Irrational

Both square roots

±22.090722

Prime factorization

2³ × 61

Cube root

7.872994

How to simplify √488

Look for the largest perfect square that divides 488. Here it is 4 (2²), because 488 = 4 × 122 and 122 has no square factor left:

The prime factorization tells the same story: 488 = 2³ × 61. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 61 stays inside.

Check: (2√122)² = 2² × 122 = 4 × 122 = 488. As a decimal, 2√122 = 2 × 11.0453610172 ≈ 22.0907220344.

Where √488 sits between perfect squares

484 = 22² and 529 = 23² are the nearest perfect squares, so √488 lies between 22 and 23. 488 is 4 above 484 and 41 below 529, so the root is closer to 22.

• Straight line between 484 and 529: 22.0889 (0.01% low)
• Tangent from 22, i.e. 22 + 4 ÷ 44: 22.0909 (0% high)
• Tangent from 23, i.e. 23 − 41 ÷ 46: 22.1087 (0.08% high)

For √488 the tangent at 22 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 488 is just 4 above 484.

Finding √488 with the Babylonian method

Picture a rectangle with an area of 488 and one side x; the other side must be 488 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √488.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √488 = 22.0907220344 to every decimal shown.

√488 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √488 the pattern is [22; 11, 44] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √488 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 − 488y² = 1. Its smallest solution in positive whole numbers is x = 243, y = 11.

√488 in geometry and everyday measurements

• A square garage floor of 488 square feet measures about 22.09 ft (22 ft 1 in) per side, and its corner-to-corner diagonal is √976 ≈ 31.2 ft.
• 488 = 2² + 22², so by the Pythagorean theorem √488 is the diagonal of a 2 × 22 rectangle — and the distance between the points (0, 0) and (2, 22) on a grid.
• Since √488 = 2√122, a length of √488 is exactly 2 copies of the length √122 laid end to end.

Square roots near √488 and related roots

• The cube root of 488 is about 7.872994.
• Because 488 = 4 × 122, the root is twice √122: 2 × 11.045361 ≈ 22.090722.

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