Square Root of 714

√714 at a glance

Exact value

√714

Decimal (10 places)

26.7207784318

Rounded

26.7 · 26.72 · 26.721

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.720778

Prime factorization

2 × 3 × 7 × 17

Cube root

8.937843

How to simplify √714

The prime factorization of 714 is 2 × 3 × 7 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √714 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 714, 2, 3, 7 and 17 appear an odd number of times, so √714 is irrational and 26.7207784318 is a rounded value.

Where √714 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √714 lies between 26 and 27. 714 is 38 above 676 and 15 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.7170 (0.01% low)
• Tangent from 26, i.e. 26 + 38 ÷ 52: 26.7308 (0.04% high)
• Tangent from 27, i.e. 27 − 15 ÷ 54: 26.7222 (0.01% high)

For √714 the tangent at 27 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 714 is just 15 below 729.

Finding √714 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 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √714 = 26.7207784318 to every decimal shown.

√714 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √714 the pattern is [26; 1, 2, 1, 1, 2, 1, 1, 2, 1, 52] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √714 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 − 714y² = 1. Its smallest solution in positive whole numbers is x = 4,115, y = 154.

√714 in geometry and everyday measurements

• 714 square feet is 66.3 m². Laid out as a square — a small house footprint or a lot — it is about 26.72 ft (26 ft 9 in) on a side.
• 714 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √714 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 13 × 23 box, because 4² + 13² + 23² = 714.

Square roots near √714 and related roots

• The cube root of 714 is about 8.937843.
• Squaring undoes the root: (√714)² = 714, while 714² = 509,796 — the number whose square root is 714.

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