Square Root of 278

√278 at a glance

Exact value

√278

Decimal (10 places)

16.6733320005

Rounded

16.7 · 16.67 · 16.673

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.673332

Prime factorization

2 × 139

Cube root

6.526519

How to simplify √278

The prime factorization of 278 is 2 × 139. Every prime appears only once, so there is no pair to bring outside the radical — √278 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 278, 2 and 139 appear an odd number of times, so √278 is irrational and 16.6733320005 is a rounded value.

Where √278 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √278 lies between 16 and 17. 278 is 22 above 256 and 11 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.6667 (0.04% low)
• Tangent from 16, i.e. 16 + 22 ÷ 32: 16.6875 (0.08% high)
• Tangent from 17, i.e. 17 − 11 ÷ 34: 16.6765 (0.02% high)

For √278 the tangent at 17 wins, missing by only 0.0031. Tangent estimates shine when the number sits close to a perfect square — here 278 is just 11 below 289.

Finding √278 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, 17 (17² = 289):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √278 = 16.6733320005 to every decimal shown.

√278 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √278 the pattern is [16; 1, 2, 16, 2, 1, 32] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √278 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 − 278y² = 1. Its smallest solution in positive whole numbers is x = 2,501, y = 150.

√278 in geometry and everyday measurements

• A square patio or deck of 278 square feet is about 16.67 ft (16 ft 8 in) on each side, so edging all the way around takes 4 × √278 ≈ 66.7 ft.
• 278 is not a sum of two whole-number squares — the prime factor 139 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √278 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 14 box, because 1² + 9² + 14² = 278.

Square roots near √278 and related roots

• The cube root of 278 is about 6.526519.
• Squaring undoes the root: (√278)² = 278, while 278² = 77,284 — the number whose square root is 278.

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