Square Root of 719

√719 at a glance

Exact value

√719

Decimal (10 places)

26.8141753556

Rounded

26.8 · 26.81 · 26.814

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.814175

Prime factorization

719

Cube root

8.958658

How to simplify √719

719 is a prime number, so its only factors are 1 and 719. There is no perfect-square factor to pull out, which means √719 is already in its simplest radical form.

The square root of any prime is irrational. If √719 were a fraction a/b in lowest terms, then a² = 719b², so 719 would divide a — and then 719 would divide b too, contradicting “lowest terms.” That is why the decimal 26.8141753556 is only a rounded value.

Where √719 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √719 lies between 26 and 27. 719 is 43 above 676 and 10 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.8113 (0.01% low)
• Tangent from 26, i.e. 26 + 43 ÷ 52: 26.8269 (0.05% high)
• Tangent from 27, i.e. 27 − 10 ÷ 54: 26.8148 (0% high)

For √719 the tangent at 27 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 719 is just 10 below 729.

Finding √719 with the Babylonian method

If a guess is too big, 719 ÷ 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√719) in one step.

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

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √719 = 26.8141753556 to every decimal shown.

√719 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √719 the pattern is [26; 1, 4, 2, 1, 1, 1, 1, 1, 4, 3, 1, 9, …] with the block of 28 terms after the semicolon repeating forever (only the first 12 of the 28 are shown). A pattern that never ends is one more proof that √719 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 − 719y² = 1. Its smallest solution in positive whole numbers is x = 403,480,310,400, y = 15,047,276,489.

√719 in geometry and everyday measurements

• 719 square feet is 66.8 m². Laid out as a square — a small house footprint or a lot — it is about 26.81 ft (26 ft 10 in) on a side.
• 719 is not a sum of two whole-number squares — 719 is itself a prime that is one less than a multiple of 4, which rules that out — so √719 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 √719 as its space diagonal.

Square roots near √719 and related roots

• The cube root of 719 is about 8.958658.
• Squaring undoes the root: (√719)² = 719, while 719² = 516,961 — the number whose square root is 719.

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