Square Root of 718

√718 at a glance

Exact value

√718

Decimal (10 places)

26.7955220139

Rounded

26.8 · 26.80 · 26.796

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.795522

Prime factorization

2 × 359

Cube root

8.954503

How to simplify √718

The prime factorization of 718 is 2 × 359. Every prime appears only once, so there is no pair to bring outside the radical — √718 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 718, 2 and 359 appear an odd number of times, so √718 is irrational and 26.7955220139 is a rounded value.

Where √718 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √718 lies between 26 and 27. 718 is 42 above 676 and 11 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.7925 (0.01% low)
• Tangent from 26, i.e. 26 + 42 ÷ 52: 26.8077 (0.05% high)
• Tangent from 27, i.e. 27 − 11 ÷ 54: 26.7963 (0% high)

For √718 the tangent at 27 wins, missing by only 0.0008. Tangent estimates shine when the number sits close to a perfect square — here 718 is just 11 below 729.

Finding √718 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, 27 (27² = 729):

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √718 = 26.7955220139 to every decimal shown.

√718 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √718 the pattern is [26; 1, 3, 1, 8, 7, 1, 1, 5, 2, 2, 1, 2, …] with the block of 40 terms after the semicolon repeating forever (only the first 12 of the 40 are shown). A pattern that never ends is one more proof that √718 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 − 718y² = 1. Its smallest solution in positive whole numbers is x = 8,933,399,183,036,079,503, y = 333,391,496,474,140,716 — 19 digits for x, even though 718 is small, which is what makes Pell’s equation famous.

√718 in geometry and everyday measurements

• 718 square feet is 66.7 m². Laid out as a square — a small house footprint or a lot — it is about 26.8 ft (26 ft 10 in) on a side.
• 718 is not a sum of two whole-number squares — the prime factor 359 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √718 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 15 × 22 box, because 3² + 15² + 22² = 718.

Square roots near √718 and related roots

• The cube root of 718 is about 8.954503.
• Squaring undoes the root: (√718)² = 718, while 718² = 515,524 — the number whose square root is 718.

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