Square Root of 428

√428 at a glance

Exact value

2√107

Decimal (10 places)

20.6881608656

Rounded

20.7 · 20.69 · 20.688

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.688161

Prime factorization

2² × 107

Cube root

7.536122

How to simplify √428

Look for the largest perfect square that divides 428. Here it is 4 (2²), because 428 = 4 × 107 and 107 has no square factor left:

The prime factorization tells the same story: 428 = 2² × 107. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 107 stays inside.

Check: (2√107)² = 2² × 107 = 4 × 107 = 428. As a decimal, 2√107 = 2 × 10.3440804328 ≈ 20.6881608656.

Where √428 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √428 lies between 20 and 21. 428 is 28 above 400 and 13 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.6829 (0.03% low)
• Tangent from 20, i.e. 20 + 28 ÷ 40: 20.7000 (0.06% high)
• Tangent from 21, i.e. 21 − 13 ÷ 42: 20.6905 (0.01% high)

For √428 the tangent at 21 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 428 is just 13 below 441.

Finding √428 with the Babylonian method

Picture a rectangle with an area of 428 and one side x; the other side must be 428 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √428.

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 √428 = 20.6881608656 to every decimal shown.

√428 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √428 the pattern is [20; 1, 2, 4, 1, 5, 10, 5, 1, 4, 2, 1, 40] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √428 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 − 428y² = 1. Its smallest solution in positive whole numbers is x = 1,850,887, y = 89,466.

√428 in geometry and everyday measurements

• A square garage floor of 428 square feet measures about 20.69 ft (20 ft 8 in) per side, and its corner-to-corner diagonal is √856 ≈ 29.3 ft.
• 428 is not a sum of two whole-number squares — the prime factor 107 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √428 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 10 × 18 box, because 2² + 10² + 18² = 428.
• Since √428 = 2√107, a length of √428 is exactly 2 copies of the length √107 laid end to end.

Square roots near √428 and related roots

• The cube root of 428 is about 7.536122.
• Because 428 = 4 × 107, the root is twice √107: 2 × 10.34408 ≈ 20.688161.

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