Square Root of 384

√384 at a glance

Exact value

8√6

Decimal (10 places)

19.5959179423

Rounded

19.6 · 19.60 · 19.596

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.595918

Prime factorization

2⁷ × 3

Cube root

7.268482

How to simplify √384

Look for the largest perfect square that divides 384. Here it is 64 (8²), because 384 = 64 × 6 and 6 has no square factor left:

The prime factorization tells the same story: 384 = 2⁷ × 3. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 2 × 3 stays inside.

384 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √384 = 2√96, and √96 can be simplified again. Using 64 straight away finishes in one step.

Check: (8√6)² = 8² × 6 = 64 × 6 = 384. As a decimal, 8√6 = 8 × 2.4494897428 ≈ 19.5959179423.

Where √384 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √384 lies between 19 and 20. 384 is 23 above 361 and 16 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.5897 (0.03% low)
• Tangent from 19, i.e. 19 + 23 ÷ 38: 19.6053 (0.05% high)
• Tangent from 20, i.e. 20 − 16 ÷ 40: 19.6000 (0.02% high)

For √384 the tangent at 20 wins, missing by only 0.0041. Tangent estimates shine when the number sits close to a perfect square — here 384 is just 16 below 400.

Finding √384 with the Babylonian method

Picture a rectangle with an area of 384 and one side x; the other side must be 384 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √384.

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 √384 = 19.5959179423 to every decimal shown.

√384 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √384 the pattern is [19; 1, 1, 2, 9, 2, 1, 1, 38] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √384 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 − 384y² = 1. Its smallest solution in positive whole numbers is x = 4,801, y = 245.

√384 in geometry and everyday measurements

• A square patio or deck of 384 square feet is about 19.6 ft (19 ft 7 in) on each side, so edging all the way around takes 4 × √384 ≈ 78.4 ft.
• 384 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √384 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 8 × 16 box, because 8² + 8² + 16² = 384.
• Since √384 = 8√6, a length of √384 is exactly 8 copies of the length √6 laid end to end.

Square roots near √384 and related roots

• The cube root of 384 is about 7.268482.
• Because 384 = 4 × 96, the root is twice √96: 2 × 9.797959 ≈ 19.595918.

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