Square Root of 423

√423 at a glance

Exact value

3√47

Decimal (10 places)

20.5669638012

Rounded

20.6 · 20.57 · 20.567

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.566964

Prime factorization

3² × 47

Cube root

7.506661

How to simplify √423

Look for the largest perfect square that divides 423. Here it is 9 (3²), because 423 = 9 × 47 and 47 has no square factor left:

The prime factorization tells the same story: 423 = 3² × 47. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 47 stays inside.

Check: (3√47)² = 3² × 47 = 9 × 47 = 423. As a decimal, 3√47 = 3 × 6.8556546004 ≈ 20.5669638012.

Where √423 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √423 lies between 20 and 21. 423 is 23 above 400 and 18 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.5610 (0.03% low)
• Tangent from 20, i.e. 20 + 23 ÷ 40: 20.5750 (0.04% high)
• Tangent from 21, i.e. 21 − 18 ÷ 42: 20.5714 (0.02% high)

For √423 the tangent at 21 wins, missing by only 0.0045. Tangent estimates shine when the number sits close to a perfect square — here 423 is just 18 below 441.

Finding √423 with the Babylonian method

If a guess is too big, 423 ÷ 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√423) in one step.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √423 = 20.5669638012 to every decimal shown.

√423 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √423 the pattern is [20; 1, 1, 3, 4, 3, 1, 1, 40] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √423 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 − 423y² = 1. Its smallest solution in positive whole numbers is x = 4,607, y = 224.

√423 in geometry and everyday measurements

• A square garage floor of 423 square feet measures about 20.57 ft (20 ft 7 in) per side, and its corner-to-corner diagonal is √846 ≈ 29.1 ft.
• 423 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √423 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √423 as its space diagonal.
• Since √423 = 3√47, a length of √423 is exactly 3 copies of the length √47 laid end to end.

Square roots near √423 and related roots

• The cube root of 423 is about 7.506661.
• Squaring undoes the root: (√423)² = 423, while 423² = 178,929 — the number whose square root is 423.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.