Square Root of 442

√442 at a glance

Exact value

√442

Decimal (10 places)

21.0237960416

Rounded

21.0 · 21.02 · 21.024

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.023796

Prime factorization

2 × 13 × 17

Cube root

7.617412

How to simplify √442

The prime factorization of 442 is 2 × 13 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √442 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 442, 2, 13 and 17 appear an odd number of times, so √442 is irrational and 21.0237960416 is a rounded value.

Where √442 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √442 lies between 21 and 22. 442 is 1 above 441 and 42 below 484, so the root is closer to 21.

• Straight line between 441 and 484: 21.0233 (0% low)
• Tangent from 21, i.e. 21 + 1 ÷ 42: 21.0238 (0% high)
• Tangent from 22, i.e. 22 − 42 ÷ 44: 21.0455 (0.1% high)

For √442 the tangent at 21 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 442 is just 1 above 441.

Finding √442 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

Start from the nearest whole number, 21 (21² = 441):

Because the starting guess was already close, two steps are enough to match √442 = 21.0237960416 to every decimal shown.

√442 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √442 the pattern is [21; 42] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 442 is one more than a perfect square (21² + 1). A pattern that never ends is one more proof that √442 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 − 442y² = 1. Its smallest solution in positive whole numbers is x = 883, y = 42. Because the period is odd, the equation with −1 on the right also has a solution: 21² − 442 × 1² = −1.

√442 in geometry and everyday measurements

• A square garage floor of 442 square feet measures about 21.02 ft (21 ft) per side, and its corner-to-corner diagonal is √884 ≈ 29.7 ft.
• 442 = 1² + 21² = 9² + 19², so by the Pythagorean theorem √442 is the diagonal of rectangles measuring 1 × 21 and 9 × 19 — and the distance between the points (0, 0) and (1, 21) on a grid.

Square roots near √442 and related roots

• The cube root of 442 is about 7.617412.
• Squaring undoes the root: (√442)² = 442, while 442² = 195,364 — the number whose square root is 442.

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