Square Root of 688

√688 at a glance

Exact value

4√43

Decimal (10 places)

26.2297540972

Rounded

26.2 · 26.23 · 26.230

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.229754

Prime factorization

2⁴ × 43

Cube root

8.828010

How to simplify √688

Look for the largest perfect square that divides 688. Here it is 16 (4²), because 688 = 16 × 43 and 43 has no square factor left:

The prime factorization tells the same story: 688 = 2⁴ × 43. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 43 stays inside.

688 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √688 = 2√172, and √172 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√43)² = 4² × 43 = 16 × 43 = 688. As a decimal, 4√43 = 4 × 6.5574385243 ≈ 26.2297540972.

Where √688 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √688 lies between 26 and 27. 688 is 12 above 676 and 41 below 729, so the root is closer to 26.

• Straight line between 676 and 729: 26.2264 (0.01% low)
• Tangent from 26, i.e. 26 + 12 ÷ 52: 26.2308 (0% high)
• Tangent from 27, i.e. 27 − 41 ÷ 54: 26.2407 (0.04% high)

For √688 the tangent at 26 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 688 is just 12 above 676.

Finding √688 with the Babylonian method

Picture a rectangle with an area of 688 and one side x; the other side must be 688 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √688.

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 √688 = 26.2297540972 to every decimal shown.

√688 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √688 the pattern is [26; 4, 2, 1, 5, 7, 3, 7, 5, 1, 2, 4, 52] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √688 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 − 688y² = 1. Its smallest solution in positive whole numbers is x = 24,248,647, y = 924,471.

√688 in geometry and everyday measurements

• 688 square feet is 63.9 m². Laid out as a square — a small house footprint or a lot — it is about 26.23 ft (26 ft 3 in) on a side.
• 688 is not a sum of two whole-number squares — the prime factor 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √688 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 12 × 12 × 20 box, because 12² + 12² + 20² = 688.
• Since √688 = 4√43, a length of √688 is exactly 4 copies of the length √43 laid end to end.

Square roots near √688 and related roots

• The cube root of 688 is about 8.828010.
• Because 688 = 4 × 172, the root is twice √172: 2 × 13.114877 ≈ 26.229754.

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