Square Root of 696

√696 at a glance

Exact value

2√174

Decimal (10 places)

26.3818119165

Rounded

26.4 · 26.38 · 26.382

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.381812

Prime factorization

2³ × 3 × 29

Cube root

8.862095

How to simplify √696

Look for the largest perfect square that divides 696. Here it is 4 (2²), because 696 = 4 × 174 and 174 has no square factor left:

The prime factorization tells the same story: 696 = 2³ × 3 × 29. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 29 stays inside.

Check: (2√174)² = 2² × 174 = 4 × 174 = 696. As a decimal, 2√174 = 2 × 13.1909059583 ≈ 26.3818119165.

Where √696 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √696 lies between 26 and 27. 696 is 20 above 676 and 33 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.3774 (0.02% low)
• Tangent from 26, i.e. 26 + 20 ÷ 52: 26.3846 (0.01% high)
• Tangent from 27, i.e. 27 − 33 ÷ 54: 26.3889 (0.03% high)

For √696 the tangent at 26 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 696 is just 20 above 676.

Finding √696 with the Babylonian method

Picture a rectangle with an area of 696 and one side x; the other side must be 696 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √696.

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 √696 = 26.3818119165 to every decimal shown.

√696 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √696 the pattern is [26; 2, 1, 1, 1, 1, 1, 2, 52] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √696 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 − 696y² = 1. Its smallest solution in positive whole numbers is x = 1,451, y = 55.

√696 in geometry and everyday measurements

• 696 square feet is 64.7 m². Laid out as a square — a small house footprint or a lot — it is about 26.38 ft (26 ft 5 in) on a side.
• 696 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √696 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 26 box, because 2² + 4² + 26² = 696.
• Since √696 = 2√174, a length of √696 is exactly 2 copies of the length √174 laid end to end.

Square roots near √696 and related roots

• The cube root of 696 is about 8.862095.
• Because 696 = 4 × 174, the root is twice √174: 2 × 13.190906 ≈ 26.381812.

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