Square Root of 164

√164 at a glance

Exact value

2√41

Decimal (10 places)

12.8062484749

Rounded

12.8 · 12.81 · 12.806

Perfect square?

No — between 12² and 13²

Rational?

Irrational

Both square roots

±12.806248

Prime factorization

2² × 41

Cube root

5.473704

How to simplify √164

Look for the largest perfect square that divides 164. Here it is 4 (2²), because 164 = 4 × 41 and 41 has no square factor left:

The prime factorization tells the same story: 164 = 2² × 41. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 41 stays inside.

Check: (2√41)² = 2² × 41 = 4 × 41 = 164. As a decimal, 2√41 = 2 × 6.4031242374 ≈ 12.8062484749.

Where √164 sits between perfect squares

144 = 12² and 169 = 13² are the nearest perfect squares, so √164 lies between 12 and 13. 164 is 20 above 144 and 5 below 169, so the root is closer to 13.

• Straight line between 144 and 169: 12.8000 (0.05% low)
• Tangent from 12, i.e. 12 + 20 ÷ 24: 12.8333 (0.21% high)
• Tangent from 13, i.e. 13 − 5 ÷ 26: 12.8077 (0.01% high)

For √164 the tangent at 13 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 164 is just 5 below 169.

Finding √164 with the Babylonian method

Picture a rectangle with an area of 164 and one side x; the other side must be 164 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √164.

Start from the nearest whole number, 13 (13² = 169):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √164 = 12.8062484749 to every decimal shown.

√164 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √164 the pattern is [12; 1, 4, 6, 4, 1, 24] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √164 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 − 164y² = 1. Its smallest solution in positive whole numbers is x = 2,049, y = 160.

√164 in geometry and everyday measurements

• A square room or garden bed covering 164 square feet measures about 12.81 ft (12 ft 10 in) along each wall.
• 164 = 8² + 10², so by the Pythagorean theorem √164 is the diagonal of a 8 × 10 rectangle — and the distance between the points (0, 0) and (8, 10) on a grid.
• Since √164 = 2√41, a length of √164 is exactly 2 copies of the length √41 laid end to end.

Square roots near √164 and related roots

• The cube root of 164 is about 5.473704.
• Four times the radicand doubles the root: √656 = 2 × √164 ≈ 25.612497.

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