Square Root of 763

√763 at a glance

Exact value

√763

Decimal (10 places)

27.6224546339

Rounded

27.6 · 27.62 · 27.622

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.622455

Prime factorization

7 × 109

Cube root

9.137797

How to simplify √763

The prime factorization of 763 is 7 × 109. Every prime appears only once, so there is no pair to bring outside the radical — √763 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 763, 7 and 109 appear an odd number of times, so √763 is irrational and 27.6224546339 is a rounded value.

Where √763 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √763 lies between 27 and 28. 763 is 34 above 729 and 21 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.6182 (0.02% low)
• Tangent from 27, i.e. 27 + 34 ÷ 54: 27.6296 (0.03% high)
• Tangent from 28, i.e. 28 − 21 ÷ 56: 27.6250 (0.01% high)

For √763 the tangent at 28 wins, missing by only 0.0025. Tangent estimates shine when the number sits close to a perfect square — here 763 is just 21 below 784.

Finding √763 with the Babylonian method

If a guess is too big, 763 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√763) in one step.

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 √763 = 27.6224546339 to every decimal shown.

√763 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √763 the pattern is [27; 1, 1, 1, 1, 1, 5, 1, 1, 17, 1, 6, 1, …] with the block of 22 terms after the semicolon repeating forever (only the first 12 of the 22 are shown). A pattern that never ends is one more proof that √763 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 − 763y² = 1. Its smallest solution in positive whole numbers is x = 719,724,601, y = 26,055,780.

√763 in geometry and everyday measurements

• 763 square feet is 70.9 m². Laid out as a square — a small house footprint or a lot — it is about 27.62 ft (27 ft 7 in) on a side.
• 763 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 √763 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 5 × 27 box, because 3² + 5² + 27² = 763.

Square roots near √763 and related roots

• The cube root of 763 is about 9.137797.
• Squaring undoes the root: (√763)² = 763, while 763² = 582,169 — the number whose square root is 763.

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