Square Root of 761

√761 at a glance

Exact value

√761

Decimal (10 places)

27.5862284483

Rounded

27.6 · 27.59 · 27.586

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.586228

Prime factorization

761

Cube root

9.129806

How to simplify √761

761 is a prime number, so its only factors are 1 and 761. There is no perfect-square factor to pull out, which means √761 is already in its simplest radical form.

The square root of any prime is irrational. If √761 were a fraction a/b in lowest terms, then a² = 761b², so 761 would divide a — and then 761 would divide b too, contradicting “lowest terms.” That is why the decimal 27.5862284483 is only a rounded value.

Where √761 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √761 lies between 27 and 28. 761 is 32 above 729 and 23 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.5818 (0.02% low)
• Tangent from 27, i.e. 27 + 32 ÷ 54: 27.5926 (0.02% high)
• Tangent from 28, i.e. 28 − 23 ÷ 56: 27.5893 (0.01% high)

For √761 the tangent at 28 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 761 is just 23 below 784.

Finding √761 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 761: following the tangent line down to zero simplifies to averaging x with 761 ÷ x.

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 √761 = 27.5862284483 to every decimal shown.

√761 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √761 the pattern is [27; 1, 1, 2, 2, 1, 1, 54] with the block of 7 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √761 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 − 761y² = 1. Its smallest solution in positive whole numbers is x = 1,280,001, y = 46,400. Because the period is odd, the equation with −1 on the right also has a solution: 800² − 761 × 29² = −1.

√761 in geometry and everyday measurements

• 761 square feet is 70.7 m². Laid out as a square — a small house footprint or a lot — it is about 27.59 ft (27 ft 7 in) on a side.
• 761 = 19² + 20², so by the Pythagorean theorem √761 is the diagonal of a 19 × 20 rectangle — and the distance between the points (0, 0) and (19, 20) on a grid.

Square roots near √761 and related roots

• The cube root of 761 is about 9.129806.
• Squaring undoes the root: (√761)² = 761, while 761² = 579,121 — the number whose square root is 761.

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