Square Root of 728

√728 at a glance

Exact value

2√182

Decimal (10 places)

26.9814751265

Rounded

27.0 · 26.98 · 26.981

Perfect square?

No — between 26² and 27²

Rational?

Irrational

Both square roots

±26.981475

Prime factorization

2³ × 7 × 13

Cube root

8.995883

How to simplify √728

Look for the largest perfect square that divides 728. Here it is 4 (2²), because 728 = 4 × 182 and 182 has no square factor left:

The prime factorization tells the same story: 728 = 2³ × 7 × 13. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 7 × 13 stays inside.

Check: (2√182)² = 2² × 182 = 4 × 182 = 728. As a decimal, 2√182 = 2 × 13.4907375632 ≈ 26.9814751265.

Where √728 sits between perfect squares

676 = 26² and 729 = 27² are the nearest perfect squares, so √728 lies between 26 and 27. 728 is 52 above 676 and 1 below 729, so the root is closer to 27.

• Straight line between 676 and 729: 26.9811 (0% low)
• Tangent from 26, i.e. 26 + 52 ÷ 52: 27.0000 (0.07% high)
• Tangent from 27, i.e. 27 − 1 ÷ 54: 26.9815 (0% high)

For √728 the tangent at 27 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 728 is just 1 below 729.

Finding √728 with the Babylonian method

Picture a rectangle with an area of 728 and one side x; the other side must be 728 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √728.

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

Because the starting guess was already close, two steps are enough to match √728 = 26.9814751265 to every decimal shown.

√728 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √728 the pattern is [26; 1, 52] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √728 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 − 728y² = 1. Its smallest solution in positive whole numbers is x = 27, y = 1.

√728 in geometry and everyday measurements

• 728 square feet is 67.6 m². Laid out as a square — a small house footprint or a lot — it is about 26.98 ft (27 ft) on a side.
• 728 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √728 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 18 × 20 box, because 2² + 18² + 20² = 728.
• Since √728 = 2√182, a length of √728 is exactly 2 copies of the length √182 laid end to end.

Square roots near √728 and related roots

• The cube root of 728 is about 8.995883.
• Because 728 = 4 × 182, the root is twice √182: 2 × 13.490738 ≈ 26.981475.

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