Square Root of 668

√668 at a glance

Exact value

2√167

Decimal (10 places)

25.8456959666

Rounded

25.8 · 25.85 · 25.846

Perfect square?

No — between 25² and 26²

Rational?

Irrational

Both square roots

±25.845696

Prime factorization

2² × 167

Cube root

8.741625

How to simplify √668

Look for the largest perfect square that divides 668. Here it is 4 (2²), because 668 = 4 × 167 and 167 has no square factor left:

The prime factorization tells the same story: 668 = 2² × 167. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 167 stays inside.

Check: (2√167)² = 2² × 167 = 4 × 167 = 668. As a decimal, 2√167 = 2 × 12.9228479833 ≈ 25.8456959666.

Where √668 sits between perfect squares

625 = 25² and 676 = 26² are the nearest perfect squares, so √668 lies between 25 and 26. 668 is 43 above 625 and 8 below 676, so the root is closer to 26.

• Straight line between 625 and 676: 25.8431 (0.01% low)
• Tangent from 25, i.e. 25 + 43 ÷ 50: 25.8600 (0.06% high)
• Tangent from 26, i.e. 26 − 8 ÷ 52: 25.8462 (0% high)

For √668 the tangent at 26 wins, missing by only 0.0005. Tangent estimates shine when the number sits close to a perfect square — here 668 is just 8 below 676.

Finding √668 with the Babylonian method

Picture a rectangle with an area of 668 and one side x; the other side must be 668 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √668.

Start from the nearest whole number, 26 (26² = 676):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √668 = 25.8456959666 to every decimal shown.

√668 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √668 the pattern is [25; 1, 5, 2, 12, 2, 5, 1, 50] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √668 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 − 668y² = 1. Its smallest solution in positive whole numbers is x = 56,447, y = 2,184.

√668 in geometry and everyday measurements

• 668 square feet is 62.1 m². Laid out as a square — a small house footprint or a lot — it is about 25.85 ft (25 ft 10 in) on a side.
• 668 is not a sum of two whole-number squares — the prime factor 167 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √668 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √668 as its space diagonal.
• Since √668 = 2√167, a length of √668 is exactly 2 copies of the length √167 laid end to end.

Square roots near √668 and related roots

• The cube root of 668 is about 8.741625.
• Because 668 = 4 × 167, the root is twice √167: 2 × 12.922848 ≈ 25.845696.

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