Square Root of 768

√768 at a glance

Exact value

16√3

Decimal (10 places)

27.7128129211

Rounded

27.7 · 27.71 · 27.713

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.712813

Prime factorization

2⁸ × 3

Cube root

9.157714

How to simplify √768

Look for the largest perfect square that divides 768. Here it is 256 (16²), because 768 = 256 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 768 = 2⁸ × 3. Each pair of equal primes leaves the radical as one factor, so 2⁴ comes out and 3 stays inside.

768 has 4 square factors (4, 16, 64 and 256). Starting with a smaller one still works but takes more rounds: √768 = 2√192, and √192 can be simplified again. Using 256 straight away finishes in one step.

Check: (16√3)² = 16² × 3 = 256 × 3 = 768. As a decimal, 16√3 = 16 × 1.7320508076 ≈ 27.7128129211.

Where √768 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √768 lies between 27 and 28. 768 is 39 above 729 and 16 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.7091 (0.01% low)
• Tangent from 27, i.e. 27 + 39 ÷ 54: 27.7222 (0.03% high)
• Tangent from 28, i.e. 28 − 16 ÷ 56: 27.7143 (0.01% high)

For √768 the tangent at 28 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 768 is just 16 below 784.

Finding √768 with the Babylonian method

Picture a rectangle with an area of 768 and one side x; the other side must be 768 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √768.

Start from the nearest whole number, 28 (28² = 784):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √768 = 27.7128129211 to every decimal shown.

√768 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √768 the pattern is [27; 1, 2, 2, 13, 2, 2, 1, 54] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √768 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 − 768y² = 1. Its smallest solution in positive whole numbers is x = 18,817, y = 679.

√768 in geometry and everyday measurements

• 768 square feet is 71.3 m². Laid out as a square — a small house footprint or a lot — it is about 27.71 ft (27 ft 9 in) on a side.
• 768 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 √768 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 16 × 16 × 16 box, because 16² + 16² + 16² = 768.
• Since √768 = 16√3, a length of √768 is exactly 16 copies of the length √3 laid end to end.

Square roots near √768 and related roots

• The cube root of 768 is about 9.157714.
• Because 768 = 4 × 192, the root is twice √192: 2 × 13.856406 ≈ 27.712813.

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