Square Root of 392

√392 at a glance

Exact value

14√2

Decimal (10 places)

19.7989898732

Rounded

19.8 · 19.80 · 19.799

Perfect square?

No — between 19² and 20²

Rational?

Irrational

Both square roots

±19.798990

Prime factorization

2³ × 7²

Cube root

7.318611

How to simplify √392

Look for the largest perfect square that divides 392. Here it is 196 (14²), because 392 = 196 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 392 = 2³ × 7². Each pair of equal primes leaves the radical as one factor, so 2 × 7 comes out and 2 stays inside.

392 has 3 square factors (4, 49 and 196). Starting with a smaller one still works but takes more rounds: √392 = 2√98, and √98 can be simplified again. Using 196 straight away finishes in one step.

Check: (14√2)² = 14² × 2 = 196 × 2 = 392. As a decimal, 14√2 = 14 × 1.4142135624 ≈ 19.7989898732.

Where √392 sits between perfect squares

361 = 19² and 400 = 20² are the nearest perfect squares, so √392 lies between 19 and 20. 392 is 31 above 361 and 8 below 400, so the root is closer to 20.

• Straight line between 361 and 400: 19.7949 (0.02% low)
• Tangent from 19, i.e. 19 + 31 ÷ 38: 19.8158 (0.08% high)
• Tangent from 20, i.e. 20 − 8 ÷ 40: 19.8000 (0.01% high)

For √392 the tangent at 20 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 392 is just 8 below 400.

Finding √392 with the Babylonian method

Picture a rectangle with an area of 392 and one side x; the other side must be 392 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √392.

Start from the nearest whole number, 20 (20² = 400):

The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √392 = 19.7989898732 to every decimal shown.

√392 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √392 the pattern is [19; 1, 3, 1, 38] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √392 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 − 392y² = 1. Its smallest solution in positive whole numbers is x = 99, y = 5.

√392 in geometry and everyday measurements

• A square patio or deck of 392 square feet is about 19.8 ft (19 ft 10 in) on each side, so edging all the way around takes 4 × √392 ≈ 79.2 ft.
• 392 = 14² + 14², so by the Pythagorean theorem √392 is the diagonal of a 14 × 14 rectangle — and the distance between the points (0, 0) and (14, 14) on a grid.
• Since √392 = 14√2, a length of √392 is exactly 14 copies of the length √2 laid end to end.

Square roots near √392 and related roots

• The cube root of 392 is about 7.318611.
• Because 392 = 4 × 98, the root is twice √98: 2 × 9.899495 ≈ 19.79899.

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