Square Root of 192

√192 at a glance

Exact value

8√3

Decimal (10 places)

13.8564064606

Rounded

13.9 · 13.86 · 13.856

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.856406

Prime factorization

2⁶ × 3

Cube root

5.768998

How to simplify √192

Look for the largest perfect square that divides 192. Here it is 64 (8²), because 192 = 64 × 3 and 3 has no square factor left:

The prime factorization tells the same story: 192 = 2⁶ × 3. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 3 stays inside.

192 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √192 = 2√48, and √48 can be simplified again. Using 64 straight away finishes in one step.

Check: (8√3)² = 8² × 3 = 64 × 3 = 192. As a decimal, 8√3 = 8 × 1.7320508076 ≈ 13.8564064606.

Where √192 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √192 lies between 13 and 14. 192 is 23 above 169 and 4 below 196, so the root is closer to 14.

• Straight line between 169 and 196: 13.8519 (0.03% low)
• Tangent from 13, i.e. 13 + 23 ÷ 26: 13.8846 (0.2% high)
• Tangent from 14, i.e. 14 − 4 ÷ 28: 13.8571 (0.01% high)

For √192 the tangent at 14 wins, missing by only 0.0007. Tangent estimates shine when the number sits close to a perfect square — here 192 is just 4 below 196.

Finding √192 with the Babylonian method

Picture a rectangle with an area of 192 and one side x; the other side must be 192 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √192.

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

The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √192 = 13.8564064606 to every decimal shown.

√192 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √192 the pattern is [13; 1, 5, 1, 26] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √192 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 − 192y² = 1. Its smallest solution in positive whole numbers is x = 97, y = 7.

√192 in geometry and everyday measurements

• A square room or garden bed covering 192 square feet measures about 13.86 ft (13 ft 10 in) along each wall.
• 192 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √192 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 8 × 8 × 8 box, because 8² + 8² + 8² = 192.
• Since √192 = 8√3, a length of √192 is exactly 8 copies of the length √3 laid end to end.

Square roots near √192 and related roots

• The cube root of 192 is about 5.768998.
• Four times the radicand doubles the root: √768 = 2 × √192 ≈ 27.712813.

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