Square Root of 779

√779 at a glance

Exact value

√779

Decimal (10 places)

27.9105714739

Rounded

27.9 · 27.91 · 27.911

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.910571

Prime factorization

19 × 41

Cube root

9.201229

How to simplify √779

The prime factorization of 779 is 19 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √779 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 779, 19 and 41 appear an odd number of times, so √779 is irrational and 27.9105714739 is a rounded value.

Where √779 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √779 lies between 27 and 28. 779 is 50 above 729 and 5 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.9091 (0.01% low)
• Tangent from 27, i.e. 27 + 50 ÷ 54: 27.9259 (0.06% high)
• Tangent from 28, i.e. 28 − 5 ÷ 56: 27.9107 (0% high)

For √779 the tangent at 28 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 779 is just 5 below 784.

Finding √779 with the Babylonian method

If a guess is too big, 779 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√779) in one step.

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

The count of correct decimals went 3, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √779 = 27.9105714739 to every decimal shown.

√779 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √779 the pattern is [27; 1, 10, 5, 2, 27, 2, 5, 10, 1, 54] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √779 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 − 779y² = 1. Its smallest solution in positive whole numbers is x = 11,785,490, y = 422,259.

√779 in geometry and everyday measurements

• 779 square feet is 72.4 m². Laid out as a square — a small house footprint or a lot — it is about 27.91 ft (27 ft 11 in) on a side.
• 779 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √779 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 7 × 27 box, because 1² + 7² + 27² = 779.

Square roots near √779 and related roots

• The cube root of 779 is about 9.201229.
• Squaring undoes the root: (√779)² = 779, while 779² = 606,841 — the number whose square root is 779.

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