Square Root of 730

√730 at a glance

Exact value

√730

Decimal (10 places)

27.0185121722

Rounded

27.0 · 27.02 · 27.019

Perfect square?

No — between 27² and 28²

Rational?

Irrational

Both square roots

±27.018512

Prime factorization

2 × 5 × 73

Cube root

9.004113

How to simplify √730

The prime factorization of 730 is 2 × 5 × 73. Every prime appears only once, so there is no pair to bring outside the radical — √730 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 730, 2, 5 and 73 appear an odd number of times, so √730 is irrational and 27.0185121722 is a rounded value.

Where √730 sits between perfect squares

729 = 27² and 784 = 28² are the nearest perfect squares, so √730 lies between 27 and 28. 730 is 1 above 729 and 54 below 784, so the root is closer to 27.

• Straight line between 729 and 784: 27.0182 (0% low)
• Tangent from 27, i.e. 27 + 1 ÷ 54: 27.0185 (0% high)
• Tangent from 28, i.e. 28 − 54 ÷ 56: 27.0357 (0.06% high)

For √730 the tangent at 27 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 730 is just 1 above 729.

Finding √730 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):

Because the starting guess was already close, two steps are enough to match √730 = 27.0185121722 to every decimal shown.

√730 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √730 the pattern is [27; 54] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 730 is one more than a perfect square (27² + 1). A pattern that never ends is one more proof that √730 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 − 730y² = 1. Its smallest solution in positive whole numbers is x = 1,459, y = 54. Because the period is odd, the equation with −1 on the right also has a solution: 27² − 730 × 1² = −1.

√730 in geometry and everyday measurements

• 730 square feet is 67.8 m². Laid out as a square — a small house footprint or a lot — it is about 27.02 ft (27 ft) on a side.
• 730 = 1² + 27² = 17² + 21², so by the Pythagorean theorem √730 is the diagonal of rectangles measuring 1 × 27 and 17 × 21 — and the distance between the points (0, 0) and (1, 27) on a grid.

Square roots near √730 and related roots

• The cube root of 730 is about 9.004113.
• Squaring undoes the root: (√730)² = 730, while 730² = 532,900 — the number whose square root is 730.

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