Square Root of 760

√760 at a glance

Exact value

2√190

Decimal (10 places)

27.5680975042

Rounded

27.6 · 27.57 · 27.568

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.568098

Prime factorization

2³ × 5 × 19

Cube root

9.125805

How to simplify √760

Look for the largest perfect square that divides 760. Here it is 4 (2²), because 760 = 4 × 190 and 190 has no square factor left:

The prime factorization tells the same story: 760 = 2³ × 5 × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 5 × 19 stays inside.

Check: (2√190)² = 2² × 190 = 4 × 190 = 760. As a decimal, 2√190 = 2 × 13.7840487521 ≈ 27.5680975042.

Where √760 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √760 lies between 27 and 28. 760 is 31 above 729 and 24 below 784, so the root is closer to 28.

• Straight line between 729 and 784: 27.5636 (0.02% low)
• Tangent from 27, i.e. 27 + 31 ÷ 54: 27.5741 (0.02% high)
• Tangent from 28, i.e. 28 − 24 ÷ 56: 27.5714 (0.01% high)

For √760 the tangent at 28 wins, missing by only 0.0033. Tangent estimates shine when the number sits close to a perfect square — here 760 is just 24 below 784.

Finding √760 with the Babylonian method

Picture a rectangle with an area of 760 and one side x; the other side must be 760 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √760.

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 √760 = 27.5680975042 to every decimal shown.

√760 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √760 the pattern is [27; 1, 1, 3, 5, 1, 5, 3, 1, 1, 54] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √760 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 − 760y² = 1. Its smallest solution in positive whole numbers is x = 52,021, y = 1,887.

√760 in geometry and everyday measurements

• 760 square feet is 70.6 m². Laid out as a square — a small house footprint or a lot — it is about 27.57 ft (27 ft 7 in) on a side.
• 760 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √760 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 18 × 20 box, because 6² + 18² + 20² = 760.
• Since √760 = 2√190, a length of √760 is exactly 2 copies of the length √190 laid end to end.

Square roots near √760 and related roots

• The cube root of 760 is about 9.125805.
• Because 760 = 4 × 190, the root is twice √190: 2 × 13.784049 ≈ 27.568098.

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