Square Root of 711

√711 at a glance

Exact value

3√79

Decimal (10 places)

26.6645832519

Rounded

26.7 · 26.66 · 26.665

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.664583

Prime factorization

3² × 79

Cube root

8.925308

How to simplify √711

Look for the largest perfect square that divides 711. Here it is 9 (3²), because 711 = 9 × 79 and 79 has no square factor left:

The prime factorization tells the same story: 711 = 3² × 79. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 79 stays inside.

Check: (3√79)² = 3² × 79 = 9 × 79 = 711. As a decimal, 3√79 = 3 × 8.8881944173 ≈ 26.6645832519.

Where √711 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √711 lies between 26 and 27. 711 is 35 above 676 and 18 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.6604 (0.02% low)
• Tangent from 26, i.e. 26 + 35 ÷ 52: 26.6731 (0.03% high)
• Tangent from 27, i.e. 27 − 18 ÷ 54: 26.6667 (0.01% high)

For √711 the tangent at 27 wins, missing by only 0.0021. Tangent estimates shine when the number sits close to a perfect square — here 711 is just 18 below 729.

Finding √711 with the Babylonian method

If a guess is too big, 711 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√711) in one step.

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

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √711 = 26.6645832519 to every decimal shown.

√711 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √711 the pattern is [26; 1, 1, 1, 52] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √711 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 − 711y² = 1. Its smallest solution in positive whole numbers is x = 80, y = 3.

√711 in geometry and everyday measurements

• 711 square feet is 66.1 m². Laid out as a square — a small house footprint or a lot — it is about 26.66 ft (26 ft 8 in) on a side.
• 711 is not a sum of two whole-number squares — the prime factor 79 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √711 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √711 as its space diagonal.
• Since √711 = 3√79, a length of √711 is exactly 3 copies of the length √79 laid end to end.

Square roots near √711 and related roots

• The cube root of 711 is about 8.925308.
• Squaring undoes the root: (√711)² = 711, while 711² = 505,521 — the number whose square root is 711.

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