Square Root of 273

√273 at a glance

Exact value

√273

Decimal (10 places)

16.5227116419

Rounded

16.5 · 16.52 · 16.523

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.522712

Prime factorization

3 × 7 × 13

Cube root

6.487154

How to simplify √273

The prime factorization of 273 is 3 × 7 × 13. Every prime appears only once, so there is no pair to bring outside the radical — √273 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 273, 3, 7 and 13 appear an odd number of times, so √273 is irrational and 16.5227116419 is a rounded value.

Where √273 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √273 lies between 16 and 17. 273 is 17 above 256 and 16 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.5152 (0.05% low)
• Tangent from 16, i.e. 16 + 17 ÷ 32: 16.5313 (0.05% high)
• Tangent from 17, i.e. 17 − 16 ÷ 34: 16.5294 (0.04% high)

For √273 the tangent at 17 wins, missing by only 0.0067. Tangent estimates shine when the number sits close to a perfect square — here 273 is just 16 below 289.

Finding √273 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 273: following the tangent line down to zero simplifies to averaging x with 273 ÷ x.

Start from the nearest whole number, 17 (17² = 289):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √273 = 16.5227116419 to every decimal shown.

√273 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √273 the pattern is [16; 1, 1, 10, 1, 1, 32] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √273 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 − 273y² = 1. Its smallest solution in positive whole numbers is x = 727, y = 44.

√273 in geometry and everyday measurements

• A square patio or deck of 273 square feet is about 16.52 ft (16 ft 6 in) on each side, so edging all the way around takes 4 × √273 ≈ 66.1 ft.
• 273 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √273 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 16 box, because 1² + 4² + 16² = 273.

Square roots near √273 and related roots

• The cube root of 273 is about 6.487154.
• Squaring undoes the root: (√273)² = 273, while 273² = 74,529 — the number whose square root is 273.

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