Square Root of 481

√481 at a glance

Exact value

√481

Decimal (10 places)

21.9317121995

Rounded

21.9 · 21.93 · 21.932

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.931712

Prime factorization

13 × 37

Cube root

7.835169

How to simplify √481

The prime factorization of 481 is 13 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √481 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 481, 13 and 37 appear an odd number of times, so √481 is irrational and 21.9317121995 is a rounded value.

Where √481 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √481 lies between 21 and 22. 481 is 40 above 441 and 3 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.9302 (0.01% low)
• Tangent from 21, i.e. 21 + 40 ÷ 42: 21.9524 (0.09% high)
• Tangent from 22, i.e. 22 − 3 ÷ 44: 21.9318 (0% high)

For √481 the tangent at 22 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 481 is just 3 below 484.

Finding √481 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 481: following the tangent line down to zero simplifies to averaging x with 481 ÷ x.

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 √481 = 21.9317121995 to every decimal shown.

√481 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √481 the pattern is [21; 1, 13, 1, 1, 1, 4, 4, 1, 1, 1, 13, 1, …] with the block of 13 terms after the semicolon repeating forever (only the first 12 of the 13 are shown). A pattern that never ends is one more proof that √481 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 − 481y² = 1. Its smallest solution in positive whole numbers is x = 1,859,131,879,201, y = 84,769,117,080 — 13 digits for x, even though 481 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 964,140² − 481 × 43,961² = −1.

√481 in geometry and everyday measurements

• A square garage floor of 481 square feet measures about 21.93 ft (21 ft 11 in) per side, and its corner-to-corner diagonal is √962 ≈ 31 ft.
• 481 = 9² + 20² = 15² + 16², so by the Pythagorean theorem √481 is the diagonal of rectangles measuring 9 × 20 and 15 × 16 — and the distance between the points (0, 0) and (9, 20) on a grid.

Square roots near √481 and related roots

• The cube root of 481 is about 7.835169.
• Squaring undoes the root: (√481)² = 481, while 481² = 231,361 — the number whose square root is 481.

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