Square Root of 320

√320 at a glance

Exact value

8√5

Decimal (10 places)

17.8885438200

Rounded

17.9 · 17.89 · 17.889

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.888544

Prime factorization

2⁶ × 5

Cube root

6.839904

How to simplify √320

Look for the largest perfect square that divides 320. Here it is 64 (8²), because 320 = 64 × 5 and 5 has no square factor left:

The prime factorization tells the same story: 320 = 2⁶ × 5. Each pair of equal primes leaves the radical as one factor, so 2³ comes out and 5 stays inside.

320 has 3 square factors (4, 16 and 64). Starting with a smaller one still works but takes more rounds: √320 = 2√80, and √80 can be simplified again. Using 64 straight away finishes in one step.

Check: (8√5)² = 8² × 5 = 64 × 5 = 320. As a decimal, 8√5 = 8 × 2.2360679775 ≈ 17.8885438200.

Where √320 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √320 lies between 17 and 18. 320 is 31 above 289 and 4 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.8857 (0.02% low)
• Tangent from 17, i.e. 17 + 31 ÷ 34: 17.9118 (0.13% high)
• Tangent from 18, i.e. 18 − 4 ÷ 36: 17.8889 (0% high)

For √320 the tangent at 18 wins, missing by only 0.0003. Tangent estimates shine when the number sits close to a perfect square — here 320 is just 4 below 324.

Finding √320 with the Babylonian method

Picture a rectangle with an area of 320 and one side x; the other side must be 320 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √320.

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 √320 = 17.8885438200 to every decimal shown.

√320 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √320 the pattern is [17; 1, 7, 1, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √320 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 − 320y² = 1. Its smallest solution in positive whole numbers is x = 161, y = 9.

√320 in geometry and everyday measurements

• A square patio or deck of 320 square feet is about 17.89 ft (17 ft 11 in) on each side, so edging all the way around takes 4 × √320 ≈ 71.6 ft.
• 320 = 8² + 16², so by the Pythagorean theorem √320 is the diagonal of a 8 × 16 rectangle — and the distance between the points (0, 0) and (8, 16) on a grid.
• Since √320 = 8√5, a length of √320 is exactly 8 copies of the length √5 laid end to end.

Square roots near √320 and related roots

• The cube root of 320 is about 6.839904.
• Because 320 = 4 × 80, the root is twice √80: 2 × 8.944272 ≈ 17.888544.

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