Square Root of 746

√746 at a glance

Exact value

√746

Decimal (10 places)

27.3130005675

Rounded

27.3 · 27.31 · 27.313

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.313001

Prime factorization

2 × 373

Cube root

9.069422

How to simplify √746

The prime factorization of 746 is 2 × 373. Every prime appears only once, so there is no pair to bring outside the radical — √746 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 746, 2 and 373 appear an odd number of times, so √746 is irrational and 27.3130005675 is a rounded value.

Where √746 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √746 lies between 27 and 28. 746 is 17 above 729 and 38 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.3091 (0.01% low)
• Tangent from 27, i.e. 27 + 17 ÷ 54: 27.3148 (0.01% high)
• Tangent from 28, i.e. 28 − 38 ÷ 56: 27.3214 (0.03% high)

For √746 the tangent at 27 wins, missing by only 0.0018. Tangent estimates shine when the number sits close to a perfect square — here 746 is just 17 above 729.

Finding √746 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, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √746 = 27.3130005675 to every decimal shown.

√746 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √746 the pattern is [27; 3, 5, 7, 1, 1, 1, 1, 1, 1, 7, 5, 3, …] with the block of 13 terms after the semicolon repeating forever (only the first 12 of the 13 are shown). A pattern that never ends is one more proof that √746 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 − 746y² = 1. Its smallest solution in positive whole numbers is x = 61,268,974,069,299, y = 2,243,216,519,470 — 14 digits for x, even though 746 is small, which is what makes Pell’s equation famous. Because the period is odd, the equation with −1 on the right also has a solution: 5,534,843² − 746 × 202,645² = −1.

√746 in geometry and everyday measurements

• 746 square feet is 69.3 m². Laid out as a square — a small house footprint or a lot — it is about 27.31 ft (27 ft 4 in) on a side.
• 746 = 11² + 25², so by the Pythagorean theorem √746 is the diagonal of a 11 × 25 rectangle — and the distance between the points (0, 0) and (11, 25) on a grid.

Square roots near √746 and related roots

• The cube root of 746 is about 9.069422.
• Squaring undoes the root: (√746)² = 746, while 746² = 556,516 — the number whose square root is 746.

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