Square Root of 416

√416 at a glance

Exact value

4√26

Decimal (10 places)

20.3960780544

Rounded

20.4 · 20.40 · 20.396

Perfect square?

No — between 20² and 21²

Rational?

Irrational

Both square roots

±20.396078

Prime factorization

2⁵ × 13

Cube root

7.465022

How to simplify √416

Look for the largest perfect square that divides 416. Here it is 16 (4²), because 416 = 16 × 26 and 26 has no square factor left:

The prime factorization tells the same story: 416 = 2⁵ × 13. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 13 stays inside.

416 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √416 = 2√104, and √104 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√26)² = 4² × 26 = 16 × 26 = 416. As a decimal, 4√26 = 4 × 5.0990195136 ≈ 20.3960780544.

Where √416 sits between perfect squares

400 = 20² and 441 = 21² are the nearest perfect squares, so √416 lies between 20 and 21. 416 is 16 above 400 and 25 below 441, so the root is closer to 20.

• Straight line between 400 and 441: 20.3902 (0.03% low)
• Tangent from 20, i.e. 20 + 16 ÷ 40: 20.4000 (0.02% high)
• Tangent from 21, i.e. 21 − 25 ÷ 42: 20.4048 (0.04% high)

For √416 the tangent at 20 wins, missing by only 0.0039. Tangent estimates shine when the number sits close to a perfect square — here 416 is just 16 above 400.

Finding √416 with the Babylonian method

Picture a rectangle with an area of 416 and one side x; the other side must be 416 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √416.

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

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √416 = 20.3960780544 to every decimal shown.

√416 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √416 the pattern is [20; 2, 1, 1, 9, 1, 1, 2, 40] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √416 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 − 416y² = 1. Its smallest solution in positive whole numbers is x = 5,201, y = 255.

√416 in geometry and everyday measurements

• A square garage floor of 416 square feet measures about 20.4 ft (20 ft 5 in) per side, and its corner-to-corner diagonal is √832 ≈ 28.8 ft.
• 416 = 4² + 20², so by the Pythagorean theorem √416 is the diagonal of a 4 × 20 rectangle — and the distance between the points (0, 0) and (4, 20) on a grid.
• Since √416 = 4√26, a length of √416 is exactly 4 copies of the length √26 laid end to end.

Square roots near √416 and related roots

• The cube root of 416 is about 7.465022.
• Because 416 = 4 × 104, the root is twice √104: 2 × 10.198039 ≈ 20.396078.

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