Square Root of 429

√429 at a glance

Exact value

√429

Decimal (10 places)

20.7123151772

Rounded

20.7 · 20.71 · 20.712

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.712315

Prime factorization

3 × 11 × 13

Cube root

7.541987

How to simplify √429

The prime factorization of 429 is 3 × 11 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √429 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 429, 3, 11 and 13 appear an odd number of times, so √429 is irrational and 20.7123151772 is a rounded value.

Where √429 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √429 lies between 20 and 21. 429 is 29 above 400 and 12 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.7073 (0.02% low)
• Tangent from 20, i.e. 20 + 29 ÷ 40: 20.7250 (0.06% high)
• Tangent from 21, i.e. 21 − 12 ÷ 42: 20.7143 (0.01% high)

For √429 the tangent at 21 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 429 is just 12 below 441.

Finding √429 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 429: following the tangent line down to zero simplifies to averaging x with 429 ÷ x.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √429 = 20.7123151772 to every decimal shown.

√429 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √429 the pattern is [20; 1, 2, 2, 9, 1, 12, 1, 9, 2, 2, 1, 40] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √429 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 − 429y² = 1. Its smallest solution in positive whole numbers is x = 1,524,095, y = 73,584.

√429 in geometry and everyday measurements

• A square garage floor of 429 square feet measures about 20.71 ft (20 ft 9 in) per side, and its corner-to-corner diagonal is √858 ≈ 29.3 ft.
• 429 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √429 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 5 × 20 box, because 2² + 5² + 20² = 429.

Square roots near √429 and related roots

• The cube root of 429 is about 7.541987.
• Squaring undoes the root: (√429)² = 429, while 429² = 184,041 — the number whose square root is 429.

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