Square Root of 778

√778 at a glance

Exact value

√778

Decimal (10 places)

27.8926513620

Rounded

27.9 · 27.89 · 27.893

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.892651

Prime factorization

2 × 389

Cube root

9.197290

How to simplify √778

The prime factorization of 778 is 2 × 389. Every prime appears only once, so there is no pair to bring outside the radical — √778 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 778, 2 and 389 appear an odd number of times, so √778 is irrational and 27.8926513620 is a rounded value.

Where √778 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √778 lies between 27 and 28. 778 is 49 above 729 and 6 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.8909 (0.01% low)
• Tangent from 27, i.e. 27 + 49 ÷ 54: 27.9074 (0.05% high)
• Tangent from 28, i.e. 28 − 6 ÷ 56: 27.8929 (0% high)

For √778 the tangent at 28 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 778 is just 6 below 784.

Finding √778 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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 √778 = 27.8926513620 to every decimal shown.

√778 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √778 the pattern is [27; 1, 8, 3, 5, 1, 7, 7, 1, 5, 3, 8, 1, …] with the block of 13 terms after the semicolon repeating forever (only the first 12 of the 13 are shown). A pattern that never ends is one more proof that √778 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 − 778y² = 1. Its smallest solution in positive whole numbers is x = 5,964,562,960,504,723, y = 213,839,942,395,674 — 16 digits for x, even though 778 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 54,610,269² − 778 × 1,957,873² = −1.

√778 in geometry and everyday measurements

• 778 square feet is 72.3 m². Laid out as a square — a small house footprint or a lot — it is about 27.89 ft (27 ft 11 in) on a side.
• 778 = 7² + 27², so by the Pythagorean theorem √778 is the diagonal of a 7 × 27 rectangle — and the distance between the points (0, 0) and (7, 27) on a grid.

Square roots near √778 and related roots

• The cube root of 778 is about 9.197290.
• Squaring undoes the root: (√778)² = 778, while 778² = 605,284 — the number whose square root is 778.

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