Square Root of 738

√738 at a glance

Exact value

3√82

Decimal (10 places)

27.1661554144

Rounded

27.2 · 27.17 · 27.166

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.166155

Prime factorization

2 × 3² × 41

Cube root

9.036886

How to simplify √738

Look for the largest perfect square that divides 738. Here it is 9 (3²), because 738 = 9 × 82 and 82 has no square factor left:

The prime factorization tells the same story: 738 = 2 × 3² × 41. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 41 stays inside.

Check: (3√82)² = 3² × 82 = 9 × 82 = 738. As a decimal, 3√82 = 3 × 9.0553851381 ≈ 27.1661554144.

Where √738 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √738 lies between 27 and 28. 738 is 9 above 729 and 46 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.1636 (0.01% low)
• Tangent from 27, i.e. 27 + 9 ÷ 54: 27.1667 (0% high)
• Tangent from 28, i.e. 28 − 46 ÷ 56: 27.1786 (0.05% high)

For √738 the tangent at 27 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 738 is just 9 above 729.

Finding √738 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, 27 (27² = 729):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √738 = 27.1661554144 to every decimal shown.

√738 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √738 the pattern is [27; 6, 54] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √738 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 − 738y² = 1. Its smallest solution in positive whole numbers is x = 163, y = 6.

√738 in geometry and everyday measurements

• 738 square feet is 68.6 m². Laid out as a square — a small house footprint or a lot — it is about 27.17 ft (27 ft 2 in) on a side.
• 738 = 3² + 27², so by the Pythagorean theorem √738 is the diagonal of a 3 × 27 rectangle — and the distance between the points (0, 0) and (3, 27) on a grid.
• Since √738 = 3√82, a length of √738 is exactly 3 copies of the length √82 laid end to end.

Square roots near √738 and related roots

• The cube root of 738 is about 9.036886.
• Squaring undoes the root: (√738)² = 738, while 738² = 544,644 — the number whose square root is 738.

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