Square Root of 184

√184 at a glance

Exact value

2√46

Decimal (10 places)

13.5646599663

Rounded

13.6 · 13.56 · 13.565

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.564660

Prime factorization

2³ × 23

Cube root

5.687734

How to simplify √184

Look for the largest perfect square that divides 184. Here it is 4 (2²), because 184 = 4 × 46 and 46 has no square factor left:

The prime factorization tells the same story: 184 = 2³ × 23. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 23 stays inside.

Check: (2√46)² = 2² × 46 = 4 × 46 = 184. As a decimal, 2√46 = 2 × 6.7823299831 ≈ 13.5646599663.

Where √184 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √184 lies between 13 and 14. 184 is 15 above 169 and 12 below 196, so the root is closer to 14.

• Straight line between 169 and 196: 13.5556 (0.07% low)
• Tangent from 13, i.e. 13 + 15 ÷ 26: 13.5769 (0.09% high)
• Tangent from 14, i.e. 14 − 12 ÷ 28: 13.5714 (0.05% high)

For √184 the tangent at 14 wins, missing by only 0.0068. Tangent estimates shine when the number sits close to a perfect square — here 184 is just 12 below 196.

Finding √184 with the Babylonian method

Picture a rectangle with an area of 184 and one side x; the other side must be 184 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √184.

Start from the nearest whole number, 14 (14² = 196):

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √184 = 13.5646599663 to every decimal shown.

√184 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √184 the pattern is [13; 1, 1, 3, 2, 1, 2, 1, 2, 3, 1, 1, 26] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √184 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 − 184y² = 1. Its smallest solution in positive whole numbers is x = 24,335, y = 1,794.

√184 in geometry and everyday measurements

• A square room or garden bed covering 184 square feet measures about 13.56 ft (13 ft 7 in) along each wall.
• 184 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √184 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 6 × 12 box, because 2² + 6² + 12² = 184.
• Since √184 = 2√46, a length of √184 is exactly 2 copies of the length √46 laid end to end.

Square roots near √184 and related roots

• The cube root of 184 is about 5.687734.
• Four times the radicand doubles the root: √736 = 2 × √184 ≈ 27.12932.

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