Square Root of 689

√689 at a glance

Exact value

√689

Decimal (10 places)

26.2488094968

Rounded

26.2 · 26.25 · 26.249

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.248809

Prime factorization

13 × 53

Cube root

8.832285

How to simplify √689

The prime factorization of 689 is 13 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √689 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 689, 13 and 53 appear an odd number of times, so √689 is irrational and 26.2488094968 is a rounded value.

Where √689 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √689 lies between 26 and 27. 689 is 13 above 676 and 40 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.2453 (0.01% low)
• Tangent from 26, i.e. 26 + 13 ÷ 52: 26.2500 (0% high)
• Tangent from 27, i.e. 27 − 40 ÷ 54: 26.2593 (0.04% high)

For √689 the tangent at 26 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 689 is just 13 above 676.

Finding √689 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 689: following the tangent line down to zero simplifies to averaging x with 689 ÷ x.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √689 = 26.2488094968 to every decimal shown.

√689 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √689 the pattern is [26; 4, 52] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √689 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 − 689y² = 1. Its smallest solution in positive whole numbers is x = 105, y = 4.

√689 in geometry and everyday measurements

• 689 square feet is 64 m². Laid out as a square — a small house footprint or a lot — it is about 26.25 ft (26 ft 3 in) on a side.
• 689 = 8² + 25² = 17² + 20², so by the Pythagorean theorem √689 is the diagonal of rectangles measuring 8 × 25 and 17 × 20 — and the distance between the points (0, 0) and (8, 25) on a grid.

Square roots near √689 and related roots

• The cube root of 689 is about 8.832285.
• Squaring undoes the root: (√689)² = 689, while 689² = 474,721 — the number whose square root is 689.

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