Square Root of 749

√749 at a glance

Exact value

√749

Decimal (10 places)

27.3678643668

Rounded

27.4 · 27.37 · 27.368

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.367864

Prime factorization

7 × 107

Cube root

9.081563

How to simplify √749

The prime factorization of 749 is 7 × 107. Every prime appears only once, so there is no pair to bring outside the radical — √749 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 749, 7 and 107 appear an odd number of times, so √749 is irrational and 27.3678643668 is a rounded value.

Where √749 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √749 lies between 27 and 28. 749 is 20 above 729 and 35 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.3636 (0.02% low)
• Tangent from 27, i.e. 27 + 20 ÷ 54: 27.3704 (0.01% high)
• Tangent from 28, i.e. 28 − 35 ÷ 56: 27.3750 (0.03% high)

For √749 the tangent at 27 wins, missing by only 0.0025. Tangent estimates shine when the number sits close to a perfect square — here 749 is just 20 above 729.

Finding √749 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 749: following the tangent line down to zero simplifies to averaging x with 749 ÷ 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 √749 = 27.3678643668 to every decimal shown.

√749 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √749 the pattern is [27; 2, 1, 2, 1, 1, 4, 2, 1, 1, 13, 10, 1, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √749 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 − 749y² = 1. Its smallest solution in positive whole numbers is x = 1,084,616,384,895, y = 39,631,020,176 — 13 digits for x, even though 749 is small, which is what makes Pell’s equation famous.

√749 in geometry and everyday measurements

• 749 square feet is 69.6 m². Laid out as a square — a small house footprint or a lot — it is about 27.37 ft (27 ft 4 in) on a side.
• 749 is not a sum of two whole-number squares — the prime factor 7 and 107 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √749 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 4 × 27 box, because 2² + 4² + 27² = 749.

Square roots near √749 and related roots

• The cube root of 749 is about 9.081563.
• Squaring undoes the root: (√749)² = 749, while 749² = 561,001 — the number whose square root is 749.

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