Square Root of 710

√710 at a glance

Exact value

√710

Decimal (10 places)

26.6458251889

Rounded

26.6 · 26.65 · 26.646

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.645825

Prime factorization

2 × 5 × 71

Cube root

8.921121

How to simplify √710

The prime factorization of 710 is 2 × 5 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √710 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 710, 2, 5 and 71 appear an odd number of times, so √710 is irrational and 26.6458251889 is a rounded value.

Where √710 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √710 lies between 26 and 27. 710 is 34 above 676 and 19 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.6415 (0.02% low)
• Tangent from 26, i.e. 26 + 34 ÷ 52: 26.6538 (0.03% high)
• Tangent from 27, i.e. 27 − 19 ÷ 54: 26.6481 (0.01% high)

For √710 the tangent at 27 wins, missing by only 0.0023. Tangent estimates shine when the number sits close to a perfect square — here 710 is just 19 below 729.

Finding √710 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, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √710 = 26.6458251889 to every decimal shown.

√710 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √710 the pattern is [26; 1, 1, 1, 4, 1, 1, 1, 52] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √710 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 − 710y² = 1. Its smallest solution in positive whole numbers is x = 1,279, y = 48.

√710 in geometry and everyday measurements

• 710 square feet is 66 m². Laid out as a square — a small house footprint or a lot — it is about 26.65 ft (26 ft 8 in) on a side.
• 710 is not a sum of two whole-number squares — the prime factor 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √710 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 15 × 22 box, because 1² + 15² + 22² = 710.

Square roots near √710 and related roots

• The cube root of 710 is about 8.921121.
• Squaring undoes the root: (√710)² = 710, while 710² = 504,100 — the number whose square root is 710.

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