Square Root of 328

√328 at a glance

Exact value

2√82

Decimal (10 places)

18.1107702763

Rounded

18.1 · 18.11 · 18.111

Perfect square?

No — between 18² and 19²

Rational?

Irrational

Both square roots

±18.110770

Prime factorization

2³ × 41

Cube root

6.896434

How to simplify √328

Look for the largest perfect square that divides 328. Here it is 4 (2²), because 328 = 4 × 82 and 82 has no square factor left:

The prime factorization tells the same story: 328 = 2³ × 41. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 41 stays inside.

Check: (2√82)² = 2² × 82 = 4 × 82 = 328. As a decimal, 2√82 = 2 × 9.0553851381 ≈ 18.1107702763.

Where √328 sits between perfect squares

324 = 18² and 361 = 19² are the nearest perfect squares, so √328 lies between 18 and 19. 328 is 4 above 324 and 33 below 361, so the root is closer to 18.

• Straight line between 324 and 361: 18.1081 (0.01% low)
• Tangent from 18, i.e. 18 + 4 ÷ 36: 18.1111 (0% high)
• Tangent from 19, i.e. 19 − 33 ÷ 38: 18.1316 (0.11% high)

For √328 the tangent at 18 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 328 is just 4 above 324.

Finding √328 with the Babylonian method

Picture a rectangle with an area of 328 and one side x; the other side must be 328 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √328.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √328 = 18.1107702763 to every decimal shown.

√328 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √328 the pattern is [18; 9, 36] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √328 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 − 328y² = 1. Its smallest solution in positive whole numbers is x = 163, y = 9.

√328 in geometry and everyday measurements

• A square patio or deck of 328 square feet is about 18.11 ft (18 ft 1 in) on each side, so edging all the way around takes 4 × √328 ≈ 72.4 ft.
• 328 = 2² + 18², so by the Pythagorean theorem √328 is the diagonal of a 2 × 18 rectangle — and the distance between the points (0, 0) and (2, 18) on a grid.
• Since √328 = 2√82, a length of √328 is exactly 2 copies of the length √82 laid end to end.

Square roots near √328 and related roots

• The cube root of 328 is about 6.896434.
• Because 328 = 4 × 82, the root is twice √82: 2 × 9.055385 ≈ 18.11077.

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