Square Root of 404

√404 at a glance

Exact value

2√101

Decimal (10 places)

20.0997512422

Rounded

20.1 · 20.10 · 20.100

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.099751

Prime factorization

2² × 101

Cube root

7.392542

How to simplify √404

Look for the largest perfect square that divides 404. Here it is 4 (2²), because 404 = 4 × 101 and 101 has no square factor left:

The prime factorization tells the same story: 404 = 2² × 101. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 101 stays inside.

Check: (2√101)² = 2² × 101 = 4 × 101 = 404. As a decimal, 2√101 = 2 × 10.0498756211 ≈ 20.0997512422.

Where √404 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √404 lies between 20 and 21. 404 is 4 above 400 and 37 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.0976 (0.01% low)
• Tangent from 20, i.e. 20 + 4 ÷ 40: 20.1000 (0% high)
• Tangent from 21, i.e. 21 − 37 ÷ 42: 20.1190 (0.1% high)

For √404 the tangent at 20 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 404 is just 4 above 400.

Finding √404 with the Babylonian method

Picture a rectangle with an area of 404 and one side x; the other side must be 404 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √404.

Start from the nearest whole number, 20 (20² = 400):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √404 = 20.0997512422 to every decimal shown.

√404 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √404 the pattern is [20; 10, 40] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √404 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 − 404y² = 1. Its smallest solution in positive whole numbers is x = 201, y = 10.

√404 in geometry and everyday measurements

• A square garage floor of 404 square feet measures about 20.1 ft (20 ft 1 in) per side, and its corner-to-corner diagonal is √808 ≈ 28.4 ft.
• 404 = 2² + 20², so by the Pythagorean theorem √404 is the diagonal of a 2 × 20 rectangle — and the distance between the points (0, 0) and (2, 20) on a grid.
• Since √404 = 2√101, a length of √404 is exactly 2 copies of the length √101 laid end to end.

Square roots near √404 and related roots

• The cube root of 404 is about 7.392542.
• Because 404 = 4 × 101, the root is twice √101: 2 × 10.049876 ≈ 20.099751.

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