Square Root of 708

√708 at a glance

Exact value

2√177

Decimal (10 places)

26.6082693913

Rounded

26.6 · 26.61 · 26.608

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.608269

Prime factorization

2² × 3 × 59

Cube root

8.912737

How to simplify √708

Look for the largest perfect square that divides 708. Here it is 4 (2²), because 708 = 4 × 177 and 177 has no square factor left:

The prime factorization tells the same story: 708 = 2² × 3 × 59. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 59 stays inside.

Check: (2√177)² = 2² × 177 = 4 × 177 = 708. As a decimal, 2√177 = 2 × 13.3041346957 ≈ 26.6082693913.

Where √708 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √708 lies between 26 and 27. 708 is 32 above 676 and 21 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.6038 (0.02% low)
• Tangent from 26, i.e. 26 + 32 ÷ 52: 26.6154 (0.03% high)
• Tangent from 27, i.e. 27 − 21 ÷ 54: 26.6111 (0.01% high)

For √708 the tangent at 27 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 708 is just 21 below 729.

Finding √708 with the Babylonian method

Picture a rectangle with an area of 708 and one side x; the other side must be 708 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √708.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √708 = 26.6082693913 to every decimal shown.

√708 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √708 the pattern is [26; 1, 1, 1, 1, 4, 4, 4, 1, 1, 1, 1, 52] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √708 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 − 708y² = 1. Its smallest solution in positive whole numbers is x = 62,423, y = 2,346.

√708 in geometry and everyday measurements

• 708 square feet is 65.8 m². Laid out as a square — a small house footprint or a lot — it is about 26.61 ft (26 ft 7 in) on a side.
• 708 is not a sum of two whole-number squares — the prime factor 3 and 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √708 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 4 × 26 box, because 4² + 4² + 26² = 708.
• Since √708 = 2√177, a length of √708 is exactly 2 copies of the length √177 laid end to end.

Square roots near √708 and related roots

• The cube root of 708 is about 8.912737.
• Because 708 = 4 × 177, the root is twice √177: 2 × 13.304135 ≈ 26.608269.

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