Square Root of 288

√288 at a glance

Exact value

12√2

Decimal (10 places)

16.9705627485

Rounded

17.0 · 16.97 · 16.971

Perfect square?

No — between 16² and 17²

Rational?

Irrational

Both square roots

±16.970563

Prime factorization

2⁵ × 3²

Cube root

6.603854

How to simplify √288

Look for the largest perfect square that divides 288. Here it is 144 (12²), because 288 = 144 × 2 and 2 has no square factor left:

The prime factorization tells the same story: 288 = 2⁵ × 3². Each pair of equal primes leaves the radical as one factor, so 2² × 3 comes out and 2 stays inside.

288 has 5 square factors (4, 9, 16, 36 and 144). Starting with a smaller one still works but takes more rounds: √288 = 2√72, and √72 can be simplified again. Using 144 straight away finishes in one step.

Check: (12√2)² = 12² × 2 = 144 × 2 = 288. As a decimal, 12√2 = 12 × 1.4142135624 ≈ 16.9705627485.

Where √288 sits between perfect squares

256 = 16² and 289 = 17² are the nearest perfect squares, so √288 lies between 16 and 17. 288 is 32 above 256 and 1 below 289, so the root is closer to 17.

• Straight line between 256 and 289: 16.9697 (0.01% low)
• Tangent from 16, i.e. 16 + 32 ÷ 32: 17.0000 (0.17% high)
• Tangent from 17, i.e. 17 − 1 ÷ 34: 16.9706 (0% high)

For √288 the tangent at 17 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 288 is just 1 below 289.

Finding √288 with the Babylonian method

Picture a rectangle with an area of 288 and one side x; the other side must be 288 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √288.

Start from the nearest whole number, 17 (17² = 289):

Because the starting guess was already close, two steps are enough to match √288 = 16.9705627485 to every decimal shown.

√288 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √288 the pattern is [16; 1, 32] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √288 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 − 288y² = 1. Its smallest solution in positive whole numbers is x = 17, y = 1.

√288 in geometry and everyday measurements

• A square patio or deck of 288 square feet is about 16.97 ft (17 ft) on each side, so edging all the way around takes 4 × √288 ≈ 67.9 ft.
• 288 = 12² + 12², so by the Pythagorean theorem √288 is the diagonal of a 12 × 12 rectangle — and the distance between the points (0, 0) and (12, 12) on a grid.
• Since √288 = 12√2, a length of √288 is exactly 12 copies of the length √2 laid end to end.

Square roots near √288 and related roots

• The cube root of 288 is about 6.603854.
• Because 288 = 4 × 72, the root is twice √72: 2 × 8.485281 ≈ 16.970563.

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