Square Root of 443

√443 at a glance

Exact value

√443

Decimal (10 places)

21.0475651798

Rounded

21.0 · 21.05 · 21.048

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.047565

Prime factorization

443

Cube root

7.623152

How to simplify √443

443 is a prime number, so its only factors are 1 and 443. There is no perfect-square factor to pull out, which means √443 is already in its simplest radical form.

The square root of any prime is irrational. If √443 were a fraction a/b in lowest terms, then a² = 443b², so 443 would divide a — and then 443 would divide b too, contradicting “lowest terms.” That is why the decimal 21.0475651798 is only a rounded value.

Where √443 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √443 lies between 21 and 22. 443 is 2 above 441 and 41 below 484, so the root is closer to 21.

• Straight line between 441 and 484: 21.0465 (0.01% low)
• Tangent from 21, i.e. 21 + 2 ÷ 42: 21.0476 (0% high)
• Tangent from 22, i.e. 22 − 41 ÷ 44: 21.0682 (0.1% high)

For √443 the tangent at 21 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 443 is just 2 above 441.

Finding √443 with the Babylonian method

If a guess is too big, 443 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√443) in one step.

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

The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √443 = 21.0475651798 to every decimal shown.

√443 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √443 the pattern is [21; 21, 42] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √443 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 − 443y² = 1. Its smallest solution in positive whole numbers is x = 442, y = 21.

√443 in geometry and everyday measurements

• A square garage floor of 443 square feet measures about 21.05 ft (21 ft 1 in) per side, and its corner-to-corner diagonal is √886 ≈ 29.8 ft.
• 443 is not a sum of two whole-number squares — 443 is itself a prime that is one less than a multiple of 4, which rules that out — so √443 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 21 box, because 1² + 1² + 21² = 443.

Square roots near √443 and related roots

• The cube root of 443 is about 7.623152.
• Squaring undoes the root: (√443)² = 443, while 443² = 196,249 — the number whose square root is 443.

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