Square Root of 469

√469 at a glance

Exact value

√469

Decimal (10 places)

21.6564078277

Rounded

21.7 · 21.66 · 21.656

Perfect square?

No — between 21² and 22²

Rational?

Irrational

Both square roots

±21.656408

Prime factorization

7 × 67

Cube root

7.769462

How to simplify √469

The prime factorization of 469 is 7 × 67. Every prime appears only once, so there is no pair to bring outside the radical — √469 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 469, 7 and 67 appear an odd number of times, so √469 is irrational and 21.6564078277 is a rounded value.

Where √469 sits between perfect squares

441 = 21² and 484 = 22² are the nearest perfect squares, so √469 lies between 21 and 22. 469 is 28 above 441 and 15 below 484, so the root is closer to 22.

• Straight line between 441 and 484: 21.6512 (0.02% low)
• Tangent from 21, i.e. 21 + 28 ÷ 42: 21.6667 (0.05% high)
• Tangent from 22, i.e. 22 − 15 ÷ 44: 21.6591 (0.01% high)

For √469 the tangent at 22 wins, missing by only 0.0027. Tangent estimates shine when the number sits close to a perfect square — here 469 is just 15 below 484.

Finding √469 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 469: following the tangent line down to zero simplifies to averaging x with 469 ÷ x.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √469 = 21.6564078277 to every decimal shown.

√469 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √469 the pattern is [21; 1, 1, 1, 10, 6, 10, 1, 1, 1, 42] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √469 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 − 469y² = 1. Its smallest solution in positive whole numbers is x = 137,215, y = 6,336.

√469 in geometry and everyday measurements

• A square garage floor of 469 square feet measures about 21.66 ft (21 ft 8 in) per side, and its corner-to-corner diagonal is √938 ≈ 30.6 ft.
• 469 is not a sum of two whole-number squares — the prime factor 7 and 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √469 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 12 × 18 box, because 1² + 12² + 18² = 469.

Square roots near √469 and related roots

• The cube root of 469 is about 7.769462.
• Squaring undoes the root: (√469)² = 469, while 469² = 219,961 — the number whose square root is 469.

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