Square Root of 772

√772 at a glance

Exact value

2√193

Decimal (10 places)

27.7848879789

Rounded

27.8 · 27.78 · 27.785

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.784888

Prime factorization

2² × 193

Cube root

9.173585

How to simplify √772

Look for the largest perfect square that divides 772. Here it is 4 (2²), because 772 = 4 × 193 and 193 has no square factor left:

The prime factorization tells the same story: 772 = 2² × 193. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 193 stays inside.

Check: (2√193)² = 2² × 193 = 4 × 193 = 772. As a decimal, 2√193 = 2 × 13.8924439894 ≈ 27.7848879789.

Where √772 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √772 lies between 27 and 28. 772 is 43 above 729 and 12 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.7818 (0.01% low)
• Tangent from 27, i.e. 27 + 43 ÷ 54: 27.7963 (0.04% high)
• Tangent from 28, i.e. 28 − 12 ÷ 56: 27.7857 (0% high)

For √772 the tangent at 28 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 772 is just 12 below 784.

Finding √772 with the Babylonian method

Picture a rectangle with an area of 772 and one side x; the other side must be 772 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √772.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √772 = 27.7848879789 to every decimal shown.

√772 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √772 the pattern is [27; 1, 3, 1, 1, 1, 5, 1, 1, 7, 2, 1, 1, …] with the block of 30 terms after the semicolon repeating forever (only the first 12 of the 30 are shown). A pattern that never ends is one more proof that √772 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 − 772y² = 1. Its smallest solution in positive whole numbers is x = 6,224,323,426,849, y = 224,018,302,020 — 13 digits for x, even though 772 is small, which is what makes Pell’s equation famous.

√772 in geometry and everyday measurements

• 772 square feet is 71.7 m². Laid out as a square — a small house footprint or a lot — it is about 27.78 ft (27 ft 9 in) on a side.
• 772 = 14² + 24², so by the Pythagorean theorem √772 is the diagonal of a 14 × 24 rectangle — and the distance between the points (0, 0) and (14, 24) on a grid.
• Since √772 = 2√193, a length of √772 is exactly 2 copies of the length √193 laid end to end.

Square roots near √772 and related roots

• The cube root of 772 is about 9.173585.
• Because 772 = 4 × 193, the root is twice √193: 2 × 13.892444 ≈ 27.784888.

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