Square Root of 732

√732 at a glance

Exact value

2√183

Decimal (10 places)

27.0554985169

Rounded

27.1 · 27.06 · 27.055

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.055499

Prime factorization

2² × 3 × 61

Cube root

9.012329

How to simplify √732

Look for the largest perfect square that divides 732. Here it is 4 (2²), because 732 = 4 × 183 and 183 has no square factor left:

The prime factorization tells the same story: 732 = 2² × 3 × 61. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 3 × 61 stays inside.

Check: (2√183)² = 2² × 183 = 4 × 183 = 732. As a decimal, 2√183 = 2 × 13.5277492585 ≈ 27.0554985169.

Where √732 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √732 lies between 27 and 28. 732 is 3 above 729 and 52 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.0545 (0% low)
• Tangent from 27, i.e. 27 + 3 ÷ 54: 27.0556 (0% high)
• Tangent from 28, i.e. 28 − 52 ÷ 56: 27.0714 (0.06% high)

For √732 the tangent at 27 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 732 is just 3 above 729.

Finding √732 with the Babylonian method

Picture a rectangle with an area of 732 and one side x; the other side must be 732 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √732.

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

The count of correct decimals went 4, 10 and all 10 over 3 steps — roughly doubling each time — until the guess matched √732 = 27.0554985169 to every decimal shown.

√732 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √732 the pattern is [27; 18, 54] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √732 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 − 732y² = 1. Its smallest solution in positive whole numbers is x = 487, y = 18.

√732 in geometry and everyday measurements

• 732 square feet is 68 m². Laid out as a square — a small house footprint or a lot — it is about 27.06 ft (27 ft 1 in) on a side.
• 732 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √732 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √732 as its space diagonal.
• Since √732 = 2√183, a length of √732 is exactly 2 copies of the length √183 laid end to end.

Square roots near √732 and related roots

• The cube root of 732 is about 9.012329.
• Because 732 = 4 × 183, the root is twice √183: 2 × 13.527749 ≈ 27.055499.

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