Square Root of 776

√776 at a glance

Exact value

2√194

Decimal (10 places)

27.8567765544

Rounded

27.9 · 27.86 · 27.857

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.856777

Prime factorization

2³ × 97

Cube root

9.189402

How to simplify √776

Look for the largest perfect square that divides 776. Here it is 4 (2²), because 776 = 4 × 194 and 194 has no square factor left:

The prime factorization tells the same story: 776 = 2³ × 97. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 97 stays inside.

Check: (2√194)² = 2² × 194 = 4 × 194 = 776. As a decimal, 2√194 = 2 × 13.9283882772 ≈ 27.8567765544.

Where √776 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √776 lies between 27 and 28. 776 is 47 above 729 and 8 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.8545 (0.01% low)
• Tangent from 27, i.e. 27 + 47 ÷ 54: 27.8704 (0.05% high)
• Tangent from 28, i.e. 28 − 8 ÷ 56: 27.8571 (0% high)

For √776 the tangent at 28 wins, missing by only 0.0004. Tangent estimates shine when the number sits close to a perfect square — here 776 is just 8 below 784.

Finding √776 with the Babylonian method

Picture a rectangle with an area of 776 and one side x; the other side must be 776 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √776.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √776 = 27.8567765544 to every decimal shown.

√776 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √776 the pattern is [27; 1, 5, 1, 54] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √776 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 − 776y² = 1. Its smallest solution in positive whole numbers is x = 195, y = 7.

√776 in geometry and everyday measurements

• 776 square feet is 72.1 m². Laid out as a square — a small house footprint or a lot — it is about 27.86 ft (27 ft 10 in) on a side.
• 776 = 10² + 26², so by the Pythagorean theorem √776 is the diagonal of a 10 × 26 rectangle — and the distance between the points (0, 0) and (10, 26) on a grid.
• Since √776 = 2√194, a length of √776 is exactly 2 copies of the length √194 laid end to end.

Square roots near √776 and related roots

• The cube root of 776 is about 9.189402.
• Because 776 = 4 × 194, the root is twice √194: 2 × 13.928388 ≈ 27.856777.

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