Square Root of 697

√697 at a glance

Exact value

√697

Decimal (10 places)

26.4007575649

Rounded

26.4 · 26.40 · 26.401

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.400758

Prime factorization

17 × 41

Cube root

8.866338

How to simplify √697

The prime factorization of 697 is 17 × 41. Every prime appears only once, so there is no pair to bring outside the radical — √697 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 697, 17 and 41 appear an odd number of times, so √697 is irrational and 26.4007575649 is a rounded value.

Where √697 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √697 lies between 26 and 27. 697 is 21 above 676 and 32 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.3962 (0.02% low)
• Tangent from 26, i.e. 26 + 21 ÷ 52: 26.4038 (0.01% high)
• Tangent from 27, i.e. 27 − 32 ÷ 54: 26.4074 (0.03% high)

For √697 the tangent at 26 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 697 is just 21 above 676.

Finding √697 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 697: following the tangent line down to zero simplifies to averaging x with 697 ÷ x.

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 √697 = 26.4007575649 to every decimal shown.

√697 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √697 the pattern is [26; 2, 2, 52] with the block of 3 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √697 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 − 697y² = 1. Its smallest solution in positive whole numbers is x = 34,849, y = 1,320. Because the period is odd, the equation with −1 on the right also has a solution: 132² − 697 × 5² = −1.

√697 in geometry and everyday measurements

• 697 square feet is 64.8 m². Laid out as a square — a small house footprint or a lot — it is about 26.4 ft (26 ft 5 in) on a side.
• 697 = 11² + 24² = 16² + 21², so by the Pythagorean theorem √697 is the diagonal of rectangles measuring 11 × 24 and 16 × 21 — and the distance between the points (0, 0) and (11, 24) on a grid.

Square roots near √697 and related roots

• The cube root of 697 is about 8.866338.
• Squaring undoes the root: (√697)² = 697, while 697² = 485,809 — the number whose square root is 697.

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