Square Root of 698

√698 at a glance

Exact value

√698

Decimal (10 places)

26.4196896272

Rounded

26.4 · 26.42 · 26.420

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.419690

Prime factorization

2 × 349

Cube root

8.870576

How to simplify √698

The prime factorization of 698 is 2 × 349. Every prime appears only once, so there is no pair to bring outside the radical — √698 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 698, 2 and 349 appear an odd number of times, so √698 is irrational and 26.4196896272 is a rounded value.

Where √698 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √698 lies between 26 and 27. 698 is 22 above 676 and 31 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.4151 (0.02% low)
• Tangent from 26, i.e. 26 + 22 ÷ 52: 26.4231 (0.01% high)
• Tangent from 27, i.e. 27 − 31 ÷ 54: 26.4259 (0.02% high)

For √698 the tangent at 26 wins, missing by only 0.0034. Tangent estimates shine when the number sits close to a perfect square — here 698 is just 22 above 676.

Finding √698 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, 26 (26² = 676):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √698 = 26.4196896272 to every decimal shown.

√698 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √698 the pattern is [26; 2, 2, 1, 1, 1, 1, 2, 2, 52] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √698 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 − 698y² = 1. Its smallest solution in positive whole numbers is x = 51,999,603, y = 1,968,214. Because the period is odd, the equation with −1 on the right also has a solution: 5,099² − 698 × 193² = −1.

√698 in geometry and everyday measurements

• 698 square feet is 64.8 m². Laid out as a square — a small house footprint or a lot — it is about 26.42 ft (26 ft 5 in) on a side.
• 698 = 13² + 23², so by the Pythagorean theorem √698 is the diagonal of a 13 × 23 rectangle — and the distance between the points (0, 0) and (13, 23) on a grid.

Square roots near √698 and related roots

• The cube root of 698 is about 8.870576.
• Squaring undoes the root: (√698)² = 698, while 698² = 487,204 — the number whose square root is 698.

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