Square Root of 44

√44 at a glance

Exact value

2√11

Decimal (10 places)

6.6332495807

Rounded

6.6 · 6.63 · 6.633

Perfect square?

No — between 6² and 7²

Rational?

Irrational

Both square roots

±6.633250

Prime factorization

2² × 11

Cube root

3.530348

How to simplify √44

Look for the largest perfect square that divides 44. Here it is 4 (2²), because 44 = 4 × 11 and 11 has no square factor left:

The prime factorization tells the same story: 44 = 2² × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 11 stays inside.

Check: (2√11)² = 2² × 11 = 4 × 11 = 44. As a decimal, 2√11 = 2 × 3.3166247904 ≈ 6.6332495807.

Where √44 sits between perfect squares

36 = 6² and 49 = 7² are the nearest perfect squares, so √44 lies between 6 and 7. 44 is 8 above 36 and 5 below 49, so the root is closer to 7.

• Straight line between 36 and 49: 6.6154 (0.27% low)
• Tangent from 6, i.e. 6 + 8 ÷ 12: 6.6667 (0.5% high)
• Tangent from 7, i.e. 7 − 5 ÷ 14: 6.6429 (0.14% high)

For √44 the tangent at 7 wins, missing by only 0.0096. Tangent estimates shine when the number sits close to a perfect square — here 44 is just 5 below 49.

Finding √44 with the Babylonian method

Picture a rectangle with an area of 44 and one side x; the other side must be 44 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √44.

Start from the nearest whole number, 7 (7² = 49):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √44 = 6.6332495807 to every decimal shown.

√44 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √44 the pattern is [6; 1, 1, 1, 2, 1, 1, 1, 12] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √44 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 − 44y² = 1. Its smallest solution in positive whole numbers is x = 199, y = 30.

√44 in geometry and everyday measurements

• A square room or garden bed covering 44 square feet measures about 6.63 ft (6 ft 8 in) along each wall.
• 44 is not a sum of two whole-number squares — the prime factor 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √44 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 6 box, because 2² + 2² + 6² = 44.
• Since √44 = 2√11, a length of √44 is exactly 2 copies of the length √11 laid end to end.

Square roots near √44 and related roots

• The cube root of 44 is about 3.530348.
• Four times the radicand doubles the root: √176 = 2 × √44 ≈ 13.266499.

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