Square Root of 424

√424 at a glance

Exact value

2√106

Decimal (10 places)

20.5912602820

Rounded

20.6 · 20.59 · 20.591

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.591260

Prime factorization

2³ × 53

Cube root

7.512572

How to simplify √424

Look for the largest perfect square that divides 424. Here it is 4 (2²), because 424 = 4 × 106 and 106 has no square factor left:

The prime factorization tells the same story: 424 = 2³ × 53. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 53 stays inside.

Check: (2√106)² = 2² × 106 = 4 × 106 = 424. As a decimal, 2√106 = 2 × 10.295630141 ≈ 20.5912602820.

Where √424 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √424 lies between 20 and 21. 424 is 24 above 400 and 17 below 441, so the root is closer to 21.

• Straight line between 400 and 441: 20.5854 (0.03% low)
• Tangent from 20, i.e. 20 + 24 ÷ 40: 20.6000 (0.04% high)
• Tangent from 21, i.e. 21 − 17 ÷ 42: 20.5952 (0.02% high)

For √424 the tangent at 21 wins, missing by only 0.004. Tangent estimates shine when the number sits close to a perfect square — here 424 is just 17 below 441.

Finding √424 with the Babylonian method

Picture a rectangle with an area of 424 and one side x; the other side must be 424 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √424.

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 √424 = 20.5912602820 to every decimal shown.

√424 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √424 the pattern is [20; 1, 1, 2, 4, 5, 1, 1, 1, 9, 1, 1, 1, …] with the block of 18 terms after the semicolon repeating forever (only the first 12 of the 18 are shown). A pattern that never ends is one more proof that √424 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 − 424y² = 1. Its smallest solution in positive whole numbers is x = 32,080,051, y = 1,557,945.

√424 in geometry and everyday measurements

• A square garage floor of 424 square feet measures about 20.59 ft (20 ft 7 in) per side, and its corner-to-corner diagonal is √848 ≈ 29.1 ft.
• 424 = 10² + 18², so by the Pythagorean theorem √424 is the diagonal of a 10 × 18 rectangle — and the distance between the points (0, 0) and (10, 18) on a grid.
• Since √424 = 2√106, a length of √424 is exactly 2 copies of the length √106 laid end to end.

Square roots near √424 and related roots

• The cube root of 424 is about 7.512572.
• Because 424 = 4 × 106, the root is twice √106: 2 × 10.29563 ≈ 20.59126.

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