Square Root of 706

√706 at a glance

Exact value

√706

Decimal (10 places)

26.5706605112

Rounded

26.6 · 26.57 · 26.571

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.570661

Prime factorization

2 × 353

Cube root

8.904337

How to simplify √706

The prime factorization of 706 is 2 × 353. Every prime appears only once, so there is no pair to bring outside the radical — √706 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 706, 2 and 353 appear an odd number of times, so √706 is irrational and 26.5706605112 is a rounded value.

Where √706 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √706 lies between 26 and 27. 706 is 30 above 676 and 23 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.5660 (0.02% low)
• Tangent from 26, i.e. 26 + 30 ÷ 52: 26.5769 (0.02% high)
• Tangent from 27, i.e. 27 − 23 ÷ 54: 26.5741 (0.01% high)

For √706 the tangent at 27 wins, missing by only 0.0034. Tangent estimates shine when the number sits close to a perfect square — here 706 is just 23 below 729.

Finding √706 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 √706 = 26.5706605112 to every decimal shown.

√706 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √706 the pattern is [26; 1, 1, 3, 26, 3, 1, 1, 52] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √706 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 − 706y² = 1. Its smallest solution in positive whole numbers is x = 34,595, y = 1,302.

√706 in geometry and everyday measurements

• 706 square feet is 65.6 m². Laid out as a square — a small house footprint or a lot — it is about 26.57 ft (26 ft 7 in) on a side.
• 706 = 9² + 25², so by the Pythagorean theorem √706 is the diagonal of a 9 × 25 rectangle — and the distance between the points (0, 0) and (9, 25) on a grid.

Square roots near √706 and related roots

• The cube root of 706 is about 8.904337.
• Squaring undoes the root: (√706)² = 706, while 706² = 498,436 — the number whose square root is 706.

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