Square Root of 736

√736 at a glance

Exact value

4√46

Decimal (10 places)

27.1293199325

Rounded

27.1 · 27.13 · 27.129

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.129320

Prime factorization

2⁵ × 23

Cube root

9.028715

How to simplify √736

Look for the largest perfect square that divides 736. Here it is 16 (4²), because 736 = 16 × 46 and 46 has no square factor left:

The prime factorization tells the same story: 736 = 2⁵ × 23. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 2 × 23 stays inside.

736 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √736 = 2√184, and √184 can be simplified again. Using 16 straight away finishes in one step.

Check: (4√46)² = 4² × 46 = 16 × 46 = 736. As a decimal, 4√46 = 4 × 6.7823299831 ≈ 27.1293199325.

Where √736 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √736 lies between 27 and 28. 736 is 7 above 729 and 48 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.1273 (0.01% low)
• Tangent from 27, i.e. 27 + 7 ÷ 54: 27.1296 (0% high)
• Tangent from 28, i.e. 28 − 48 ÷ 56: 27.1429 (0.05% high)

For √736 the tangent at 27 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 736 is just 7 above 729.

Finding √736 with the Babylonian method

Picture a rectangle with an area of 736 and one side x; the other side must be 736 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √736.

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 √736 = 27.1293199325 to every decimal shown.

√736 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √736 the pattern is [27; 7, 1, 2, 1, 2, 1, 7, 54] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √736 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 − 736y² = 1. Its smallest solution in positive whole numbers is x = 24,335, y = 897.

√736 in geometry and everyday measurements

• 736 square feet is 68.4 m². Laid out as a square — a small house footprint or a lot — it is about 27.13 ft (27 ft 2 in) on a side.
• 736 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √736 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 12 × 24 box, because 4² + 12² + 24² = 736.
• Since √736 = 4√46, a length of √736 is exactly 4 copies of the length √46 laid end to end.

Square roots near √736 and related roots

• The cube root of 736 is about 9.028715.
• Because 736 = 4 × 184, the root is twice √184: 2 × 13.56466 ≈ 27.12932.

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