Square Root of 304

√304 at a glance

Exact value

4√19

Decimal (10 places)

17.4355957742

Rounded

17.4 · 17.44 · 17.436

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.435596

Prime factorization

2⁴ × 19

Cube root

6.723951

How to simplify √304

Look for the largest perfect square that divides 304. Here it is 16 (4²), because 304 = 16 × 19 and 19 has no square factor left:

The prime factorization tells the same story: 304 = 2⁴ × 19. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 19 stays inside.

304 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √304 = 2√76, and √76 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√19)² = 4² × 19 = 16 × 19 = 304. As a decimal, 4√19 = 4 × 4.3588989435 ≈ 17.4355957742.

Where √304 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √304 lies between 17 and 18. 304 is 15 above 289 and 20 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.4286 (0.04% low)
• Tangent from 17, i.e. 17 + 15 ÷ 34: 17.4412 (0.03% high)
• Tangent from 18, i.e. 18 − 20 ÷ 36: 17.4444 (0.05% high)

For √304 the tangent at 17 wins, missing by only 0.0056. Tangent estimates shine when the number sits close to a perfect square — here 304 is just 15 above 289.

Finding √304 with the Babylonian method

Picture a rectangle with an area of 304 and one side x; the other side must be 304 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √304.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √304 = 17.4355957742 to every decimal shown.

√304 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √304 the pattern is [17; 2, 3, 2, 1, 1, 1, 1, 1, 2, 3, 2, 34] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √304 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 − 304y² = 1. Its smallest solution in positive whole numbers is x = 57,799, y = 3,315.

√304 in geometry and everyday measurements

• A square patio or deck of 304 square feet is about 17.44 ft (17 ft 5 in) on each side, so edging all the way around takes 4 × √304 ≈ 69.7 ft.
• 304 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √304 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 12 × 12 box, because 4² + 12² + 12² = 304.
• Since √304 = 4√19, a length of √304 is exactly 4 copies of the length √19 laid end to end.

Square roots near √304 and related roots

• The cube root of 304 is about 6.723951.
• Because 304 = 4 × 76, the root is twice √76: 2 × 8.717798 ≈ 17.435596.

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