Square Root of 709

√709 at a glance

Exact value

√709

Decimal (10 places)

26.6270539114

Rounded

26.6 · 26.63 · 26.627

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.627054

Prime factorization

709

Cube root

8.916931

How to simplify √709

709 is a prime number, so its only factors are 1 and 709. There is no perfect-square factor to pull out, which means √709 is already in its simplest radical form.

The square root of any prime is irrational. If √709 were a fraction a/b in lowest terms, then a² = 709b², so 709 would divide a — and then 709 would divide b too, contradicting “lowest terms.” That is why the decimal 26.6270539114 is only a rounded value.

Where √709 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √709 lies between 26 and 27. 709 is 33 above 676 and 20 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.6226 (0.02% low)
• Tangent from 26, i.e. 26 + 33 ÷ 52: 26.6346 (0.03% high)
• Tangent from 27, i.e. 27 − 20 ÷ 54: 26.6296 (0.01% high)

For √709 the tangent at 27 wins, missing by only 0.0026. Tangent estimates shine when the number sits close to a perfect square — here 709 is just 20 below 729.

Finding √709 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 709: following the tangent line down to zero simplifies to averaging x with 709 ÷ x.

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 √709 = 26.6270539114 to every decimal shown.

√709 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √709 the pattern is [26; 1, 1, 1, 2, 7, 4, 3, 3, 4, 7, 2, 1, …] with the block of 15 terms after the semicolon repeating forever (only the first 12 of the 15 are shown). A pattern that never ends is one more proof that √709 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 − 709y² = 1. Its smallest solution in positive whole numbers is x = 665,782,673,992,201, y = 25,003,993,164,540 — 15 digits for x, even though 709 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 18,245,310² − 709 × 685,217² = −1.

√709 in geometry and everyday measurements

• 709 square feet is 65.9 m². Laid out as a square — a small house footprint or a lot — it is about 26.63 ft (26 ft 8 in) on a side.
• 709 = 15² + 22², so by the Pythagorean theorem √709 is the diagonal of a 15 × 22 rectangle — and the distance between the points (0, 0) and (15, 22) on a grid.

Square roots near √709 and related roots

• The cube root of 709 is about 8.916931.
• Squaring undoes the root: (√709)² = 709, while 709² = 502,681 — the number whose square root is 709.

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