Square Root of 739

√739 at a glance

Exact value

√739

Decimal (10 places)

27.1845544381

Rounded

27.2 · 27.18 · 27.185

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.184554

Prime factorization

739

Cube root

9.040966

How to simplify √739

739 is a prime number, so its only factors are 1 and 739. There is no perfect-square factor to pull out, which means √739 is already in its simplest radical form.

The square root of any prime is irrational. If √739 were a fraction a/b in lowest terms, then a² = 739b², so 739 would divide a — and then 739 would divide b too, contradicting “lowest terms.” That is why the decimal 27.1845544381 is only a rounded value.

Where √739 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √739 lies between 27 and 28. 739 is 10 above 729 and 45 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.1818 (0.01% low)
• Tangent from 27, i.e. 27 + 10 ÷ 54: 27.1852 (0% high)
• Tangent from 28, i.e. 28 − 45 ÷ 56: 27.1964 (0.04% high)

For √739 the tangent at 27 wins, missing by only 0.0006. Tangent estimates shine when the number sits close to a perfect square — here 739 is just 10 above 729.

Finding √739 with the Babylonian method

If a guess is too big, 739 ÷ 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√739) 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 √739 = 27.1845544381 to every decimal shown.

√739 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √739 the pattern is [27; 5, 2, 2, 1, 1, 3, 3, 2, 1, 8, 2, 1, …] with the block of 46 terms after the semicolon repeating forever (only the first 12 of the 46 are shown). A pattern that never ends is one more proof that √739 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 − 739y² = 1. Its smallest solution in positive whole numbers is x = 98,015,661,073,616,742,153,890, y = 3,605,564,376,516,452,758,671 — 23 digits for x, even though 739 is small, which is what makes Pell’s equation famous.

√739 in geometry and everyday measurements

• 739 square feet is 68.7 m². Laid out as a square — a small house footprint or a lot — it is about 27.18 ft (27 ft 2 in) on a side.
• 739 is not a sum of two whole-number squares — 739 is itself a prime that is one less than a multiple of 4, which rules that out — so √739 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 27 box, because 1² + 3² + 27² = 739.

Square roots near √739 and related roots

• The cube root of 739 is about 9.040966.
• Squaring undoes the root: (√739)² = 739, while 739² = 546,121 — the number whose square root is 739.

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