Square Root of 74

√74 at a glance

Exact value

√74

Decimal (10 places)

8.6023252670

Rounded

8.6 · 8.60 · 8.602

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.602325

Prime factorization

2 × 37

Cube root

4.198336

How to simplify √74

The prime factorization of 74 is 2 × 37. Every prime appears only once, so there is no pair to bring outside the radical — √74 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 74, 2 and 37 appear an odd number of times, so √74 is irrational and 8.6023252670 is a rounded value.

Where √74 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √74 lies between 8 and 9. 74 is 10 above 64 and 7 below 81, so the root is closer to 9.

• Straight line between 64 and 81: 8.5882 (0.16% low)
• Tangent from 8, i.e. 8 + 10 ÷ 16: 8.6250 (0.26% high)
• Tangent from 9, i.e. 9 − 7 ÷ 18: 8.6111 (0.1% high)

For √74 the tangent at 9 wins, missing by only 0.0088. Tangent estimates shine when the number sits close to a perfect square — here 74 is just 7 below 81.

Finding √74 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, 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 √74 = 8.6023252670 to every decimal shown.

√74 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √74 the pattern is [8; 1, 1, 1, 1, 16] with the block of 5 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √74 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 − 74y² = 1. Its smallest solution in positive whole numbers is x = 3,699, y = 430. Because the period is odd, the equation with −1 on the right also has a solution: 43² − 74 × 5² = −1.

√74 in geometry and everyday measurements

• A square room or garden bed covering 74 square feet measures about 8.6 ft (8 ft 7 in) along each wall.
• 74 = 5² + 7², so by the Pythagorean theorem √74 is the diagonal of a 5 × 7 rectangle — and the distance between the points (0, 0) and (5, 7) on a grid.

Square roots near √74 and related roots

• The cube root of 74 is about 4.198336.
• Four times the radicand doubles the root: √296 = 2 × √74 ≈ 17.204651.

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