Square Root of 743

√743 at a glance

Exact value

√743

Decimal (10 places)

27.2580263409

Rounded

27.3 · 27.26 · 27.258

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.258026

Prime factorization

743

Cube root

9.057248

How to simplify √743

743 is a prime number, so its only factors are 1 and 743. There is no perfect-square factor to pull out, which means √743 is already in its simplest radical form.

The square root of any prime is irrational. If √743 were a fraction a/b in lowest terms, then a² = 743b², so 743 would divide a — and then 743 would divide b too, contradicting “lowest terms.” That is why the decimal 27.2580263409 is only a rounded value.

Where √743 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √743 lies between 27 and 28. 743 is 14 above 729 and 41 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.2545 (0.01% low)
• Tangent from 27, i.e. 27 + 14 ÷ 54: 27.2593 (0% high)
• Tangent from 28, i.e. 28 − 41 ÷ 56: 27.2679 (0.04% high)

For √743 the tangent at 27 wins, missing by only 0.0012. Tangent estimates shine when the number sits close to a perfect square — here 743 is just 14 above 729.

Finding √743 with the Babylonian method

If a guess is too big, 743 ÷ 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√743) 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 √743 = 27.2580263409 to every decimal shown.

√743 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √743 the pattern is [27; 3, 1, 7, 27, 7, 1, 3, 54] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √743 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 − 743y² = 1. Its smallest solution in positive whole numbers is x = 714,024, y = 26,195.

√743 in geometry and everyday measurements

• 743 square feet is 69 m². Laid out as a square — a small house footprint or a lot — it is about 27.26 ft (27 ft 3 in) on a side.
• 743 is not a sum of two whole-number squares — 743 is itself a prime that is one less than a multiple of 4, which rules that out — so √743 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 √743 as its space diagonal.

Square roots near √743 and related roots

• The cube root of 743 is about 9.057248.
• Squaring undoes the root: (√743)² = 743, while 743² = 552,049 — the number whose square root is 743.

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