Square Root of 289

√289 at a glance

Exact value

17

Decimal (10 places)

17

Rounded

17

Perfect square?

Yes — 17²

Rational?

Rational (whole number)

Both square roots

±17

Prime factorization

17²

Cube root

6.611489

How to simplify √289

Write 289 as a product of primes and halve every exponent. Because each exponent is even, nothing is left under the radical sign:

289 has 2 factor pairs (1 × 289, 17 × 17). The pair with two equal numbers, 17 × 17, is what makes it a perfect square — and why it has an odd number of factors.

Every perfect square is also a sum of consecutive odd numbers starting at 1. Here 289 = 1 + 3 + 5 + 7 + … + 33, the first 17 odd numbers.

Where √289 sits between perfect squares

289 is 17². The perfect squares on either side are 256 (16²) and 324 (18²), so the gaps are 33 below and 35 above — consecutive squares always differ by consecutive odd numbers.

Finding √289 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 289: following the tangent line down to zero simplifies to averaging x with 289 ÷ x.

To show the method at work on a perfect square, start one above the answer, at 18:

The count of correct decimals went 1, 4 and all 10 over 3 steps — roughly doubling each time — until the guess matched √289 = 17.0000000000 to every decimal shown.

√289 in geometry and everyday measurements

• A square patio or deck of 289 square feet is exactly 17 ft (17 ft) on each side, so edging all the way around takes 4 × √289 ≈ 68 ft.
• 289 = 8² + 15², so 8, 15 and 17 form a Pythagorean triple: a right triangle with legs 8 and 15 has a hypotenuse of exactly 17.

Square roots near √289 and related roots

• The cube root of 289 is about 6.611489.
• Taking the square root twice gives the fourth root: √17 ≈ 4.123106.
• Squaring undoes the root: (√289)² = 289, while 289² = 83,521 — the number whose square root is 289.

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