Square Root of 754

√754 at a glance

Exact value

√754

Decimal (10 places)

27.4590604355

Rounded

27.5 · 27.46 · 27.459

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.459060

Prime factorization

2 × 13 × 29

Cube root

9.101727

How to simplify √754

The prime factorization of 754 is 2 × 13 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √754 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 754, 2, 13 and 29 appear an odd number of times, so √754 is irrational and 27.4590604355 is a rounded value.

Where √754 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √754 lies between 27 and 28. 754 is 25 above 729 and 30 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.4545 (0.02% low)
• Tangent from 27, i.e. 27 + 25 ÷ 54: 27.4630 (0.01% high)
• Tangent from 28, i.e. 28 − 30 ÷ 56: 27.4643 (0.02% high)

For √754 the tangent at 27 wins, missing by only 0.0039. Tangent estimates shine when the number sits close to a perfect square — here 754 is just 25 above 729.

Finding √754 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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 √754 = 27.4590604355 to every decimal shown.

√754 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √754 the pattern is [27; 2, 5, 1, 1, 1, 1, 5, 2, 54] with the block of 9 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √754 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 − 754y² = 1. Its smallest solution in positive whole numbers is x = 836,977,699, y = 30,480,930. Because the period is odd, the equation with −1 on the right also has a solution: 20,457² − 754 × 745² = −1.

√754 in geometry and everyday measurements

• 754 square feet is 70 m². Laid out as a square — a small house footprint or a lot — it is about 27.46 ft (27 ft 6 in) on a side.
• 754 = 5² + 27² = 15² + 23², so by the Pythagorean theorem √754 is the diagonal of rectangles measuring 5 × 27 and 15 × 23 — and the distance between the points (0, 0) and (5, 27) on a grid.

Square roots near √754 and related roots

• The cube root of 754 is about 9.101727.
• Squaring undoes the root: (√754)² = 754, while 754² = 568,516 — the number whose square root is 754.

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