Square Root of 268

√268 at a glance

Exact value

2√67

Decimal (10 places)

16.3707055437

Rounded

16.4 · 16.37 · 16.371

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.370706

Prime factorization

2² × 67

Cube root

6.447306

How to simplify √268

Look for the largest perfect square that divides 268. Here it is 4 (2²), because 268 = 4 × 67 and 67 has no square factor left:

The prime factorization tells the same story: 268 = 2² × 67. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 67 stays inside.

Check: (2√67)² = 2² × 67 = 4 × 67 = 268. As a decimal, 2√67 = 2 × 8.1853527719 ≈ 16.3707055437.

Where √268 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √268 lies between 16 and 17. 268 is 12 above 256 and 21 below 289, so the root is closer to 16.

• Straight line between 256 and 289: 16.3636 (0.04% low)
• Tangent from 16, i.e. 16 + 12 ÷ 32: 16.3750 (0.03% high)
• Tangent from 17, i.e. 17 − 21 ÷ 34: 16.3824 (0.07% high)

For √268 the tangent at 16 wins, missing by only 0.0043. Tangent estimates shine when the number sits close to a perfect square — here 268 is just 12 above 256.

Finding √268 with the Babylonian method

Picture a rectangle with an area of 268 and one side x; the other side must be 268 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √268.

Start from the nearest whole number, 16 (16² = 256):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √268 = 16.3707055437 to every decimal shown.

√268 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √268 the pattern is [16; 2, 1, 2, 3, 3, 1, 3, 1, 10, 8, 10, 1, …] with the block of 20 terms after the semicolon repeating forever (only the first 12 of the 20 are shown). A pattern that never ends is one more proof that √268 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 − 268y² = 1. Its smallest solution in positive whole numbers is x = 4,771,081,927, y = 291,440,214.

√268 in geometry and everyday measurements

• A square patio or deck of 268 square feet is about 16.37 ft (16 ft 4 in) on each side, so edging all the way around takes 4 × √268 ≈ 65.5 ft.
• 268 is not a sum of two whole-number squares — the prime factor 67 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √268 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 6 × 6 × 14 box, because 6² + 6² + 14² = 268.
• Since √268 = 2√67, a length of √268 is exactly 2 copies of the length √67 laid end to end.

Square roots near √268 and related roots

• The cube root of 268 is about 6.447306.
• Because 268 = 4 × 67, the root is twice √67: 2 × 8.185353 ≈ 16.370706.

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