Square Root of 744

√744 at a glance

Exact value

2√186

Decimal (10 places)

27.2763633940

Rounded

27.3 · 27.28 · 27.276

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.276363

Prime factorization

2³ × 3 × 31

Cube root

9.061310

How to simplify √744

Look for the largest perfect square that divides 744. Here it is 4 (2²), because 744 = 4 × 186 and 186 has no square factor left:

The prime factorization tells the same story: 744 = 2³ × 3 × 31. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 3 × 31 stays inside.

Check: (2√186)² = 2² × 186 = 4 × 186 = 744. As a decimal, 2√186 = 2 × 13.638181697 ≈ 27.2763633940.

Where √744 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √744 lies between 27 and 28. 744 is 15 above 729 and 40 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.2727 (0.01% low)
• Tangent from 27, i.e. 27 + 15 ÷ 54: 27.2778 (0.01% high)
• Tangent from 28, i.e. 28 − 40 ÷ 56: 27.2857 (0.03% high)

For √744 the tangent at 27 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 744 is just 15 above 729.

Finding √744 with the Babylonian method

Picture a rectangle with an area of 744 and one side x; the other side must be 744 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √744.

Start from the nearest whole number, 27 (27² = 729):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √744 = 27.2763633940 to every decimal shown.

√744 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √744 the pattern is [27; 3, 1, 1, 1, 1, 1, 1, 1, 3, 54] with the block of 10 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √744 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 − 744y² = 1. Its smallest solution in positive whole numbers is x = 7,501, y = 275.

√744 in geometry and everyday measurements

• 744 square feet is 69.1 m². Laid out as a square — a small house footprint or a lot — it is about 27.28 ft (27 ft 3 in) on a side.
• 744 is not a sum of two whole-number squares — the prime factor 3 and 31 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √744 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 8 × 26 box, because 2² + 8² + 26² = 744.
• Since √744 = 2√186, a length of √744 is exactly 2 copies of the length √186 laid end to end.

Square roots near √744 and related roots

• The cube root of 744 is about 9.061310.
• Because 744 = 4 × 186, the root is twice √186: 2 × 13.638182 ≈ 27.276363.

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