Square Root of 742

√742 at a glance

Exact value

√742

Decimal (10 places)

27.2396769438

Rounded

27.2 · 27.24 · 27.240

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.239677

Prime factorization

2 × 7 × 53

Cube root

9.053183

How to simplify √742

The prime factorization of 742 is 2 × 7 × 53. Every prime appears only once, so there is no pair to bring outside the radical — √742 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 742, 2, 7 and 53 appear an odd number of times, so √742 is irrational and 27.2396769438 is a rounded value.

Where √742 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √742 lies between 27 and 28. 742 is 13 above 729 and 42 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.2364 (0.01% low)
• Tangent from 27, i.e. 27 + 13 ÷ 54: 27.2407 (0% high)
• Tangent from 28, i.e. 28 − 42 ÷ 56: 27.2500 (0.04% high)

For √742 the tangent at 27 wins, missing by only 0.0011. Tangent estimates shine when the number sits close to a perfect square — here 742 is just 13 above 729.

Finding √742 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 √742 = 27.2396769438 to every decimal shown.

√742 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √742 the pattern is [27; 4, 5, 1, 4, 8, 1, 6, 1, 8, 4, 1, 5, …] with the block of 14 terms after the semicolon repeating forever (only the first 12 of the 14 are shown). A pattern that never ends is one more proof that √742 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 − 742y² = 1. Its smallest solution in positive whole numbers is x = 263,091,151, y = 9,658,380.

√742 in geometry and everyday measurements

• 742 square feet is 68.9 m². Laid out as a square — a small house footprint or a lot — it is about 27.24 ft (27 ft 3 in) on a side.
• 742 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √742 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 27 box, because 2² + 3² + 27² = 742.

Square roots near √742 and related roots

• The cube root of 742 is about 9.053183.
• Squaring undoes the root: (√742)² = 742, while 742² = 550,564 — the number whose square root is 742.

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