Square Root of 762

√762 at a glance

Exact value

√762

Decimal (10 places)

27.6043474837

Rounded

27.6 · 27.60 · 27.604

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.604347

Prime factorization

2 × 3 × 127

Cube root

9.133803

How to simplify √762

The prime factorization of 762 is 2 × 3 × 127. Every prime appears only once, so there is no pair to bring outside the radical — √762 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 762, 2, 3 and 127 appear an odd number of times, so √762 is irrational and 27.6043474837 is a rounded value.

Where √762 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √762 lies between 27 and 28. 762 is 33 above 729 and 22 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.6000 (0.02% low)
• Tangent from 27, i.e. 27 + 33 ÷ 54: 27.6111 (0.02% high)
• Tangent from 28, i.e. 28 − 22 ÷ 56: 27.6071 (0.01% high)

For √762 the tangent at 28 wins, missing by only 0.0028. Tangent estimates shine when the number sits close to a perfect square — here 762 is just 22 below 784.

Finding √762 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, 28 (28² = 784):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √762 = 27.6043474837 to every decimal shown.

√762 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √762 the pattern is [27; 1, 1, 1, 1, 8, 1, 1, 1, 1, 54] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √762 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 − 762y² = 1. Its smallest solution in positive whole numbers is x = 6,349, y = 230.

√762 in geometry and everyday measurements

• 762 square feet is 70.8 m². Laid out as a square — a small house footprint or a lot — it is about 27.6 ft (27 ft 7 in) on a side.
• 762 is not a sum of two whole-number squares — the prime factor 3 and 127 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √762 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 19 × 20 box, because 1² + 19² + 20² = 762.

Square roots near √762 and related roots

• The cube root of 762 is about 9.133803.
• Squaring undoes the root: (√762)² = 762, while 762² = 580,644 — the number whose square root is 762.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.