Square Root of 764

√764 at a glance

Exact value

2√191

Decimal (10 places)

27.6405499222

Rounded

27.6 · 27.64 · 27.641

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.640550

Prime factorization

2² × 191

Cube root

9.141787

How to simplify √764

Look for the largest perfect square that divides 764. Here it is 4 (2²), because 764 = 4 × 191 and 191 has no square factor left:

The prime factorization tells the same story: 764 = 2² × 191. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 191 stays inside.

Check: (2√191)² = 2² × 191 = 4 × 191 = 764. As a decimal, 2√191 = 2 × 13.8202749611 ≈ 27.6405499222.

Where √764 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √764 lies between 27 and 28. 764 is 35 above 729 and 20 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.6364 (0.02% low)
• Tangent from 27, i.e. 27 + 35 ÷ 54: 27.6481 (0.03% high)
• Tangent from 28, i.e. 28 − 20 ÷ 56: 27.6429 (0.01% high)

For √764 the tangent at 28 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 764 is just 20 below 784.

Finding √764 with the Babylonian method

Picture a rectangle with an area of 764 and one side x; the other side must be 764 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √764.

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 √764 = 27.6405499222 to every decimal shown.

√764 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √764 the pattern is [27; 1, 1, 1, 3, 1, 1, 2, 2, 1, 6, 4, 1, …] with the block of 32 terms after the semicolon repeating forever (only the first 12 of the 32 are shown). A pattern that never ends is one more proof that √764 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 − 764y² = 1. Its smallest solution in positive whole numbers is x = 161,784,071,999,999, y = 5,853,142,302,000 — 15 digits for x, even though 764 is small, which is what makes Pell’s equation famous.

√764 in geometry and everyday measurements

• 764 square feet is 71 m². Laid out as a square — a small house footprint or a lot — it is about 27.64 ft (27 ft 8 in) on a side.
• 764 is not a sum of two whole-number squares — the prime factor 191 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √764 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √764 as its space diagonal.
• Since √764 = 2√191, a length of √764 is exactly 2 copies of the length √191 laid end to end.

Square roots near √764 and related roots

• The cube root of 764 is about 9.141787.
• Because 764 = 4 × 191, the root is twice √191: 2 × 13.820275 ≈ 27.64055.

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