Square Root of 76

√76 at a glance

Exact value

2√19

Decimal (10 places)

8.7177978871

Rounded

8.7 · 8.72 · 8.718

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.717798

Prime factorization

2² × 19

Cube root

4.235824

How to simplify √76

Look for the largest perfect square that divides 76. Here it is 4 (2²), because 76 = 4 × 19 and 19 has no square factor left:

The prime factorization tells the same story: 76 = 2² × 19. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 19 stays inside.

Check: (2√19)² = 2² × 19 = 4 × 19 = 76. As a decimal, 2√19 = 2 × 4.3588989435 ≈ 8.7177978871.

Where √76 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √76 lies between 8 and 9. 76 is 12 above 64 and 5 below 81, so the root is closer to 9.

• Straight line between 64 and 81: 8.7059 (0.14% low)
• Tangent from 8, i.e. 8 + 12 ÷ 16: 8.7500 (0.37% high)
• Tangent from 9, i.e. 9 − 5 ÷ 18: 8.7222 (0.05% high)

For √76 the tangent at 9 wins, missing by only 0.0044. Tangent estimates shine when the number sits close to a perfect square — here 76 is just 5 below 81.

Finding √76 with the Babylonian method

Picture a rectangle with an area of 76 and one side x; the other side must be 76 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √76.

Start from the nearest whole number, 9 (9² = 81):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √76 = 8.7177978871 to every decimal shown.

√76 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √76 the pattern is [8; 1, 2, 1, 1, 5, 4, 5, 1, 1, 2, 1, 16] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √76 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 − 76y² = 1. Its smallest solution in positive whole numbers is x = 57,799, y = 6,630.

√76 in geometry and everyday measurements

• A square room or garden bed covering 76 square feet measures about 8.72 ft (8 ft 9 in) along each wall.
• 76 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 √76 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 6 box, because 2² + 6² + 6² = 76.
• Since √76 = 2√19, a length of √76 is exactly 2 copies of the length √19 laid end to end.

Square roots near √76 and related roots

• The cube root of 76 is about 4.235824.
• Four times the radicand doubles the root: √304 = 2 × √76 ≈ 17.435596.

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