Square Root of 290

√290 at a glance

Exact value

√290

Decimal (10 places)

17.0293863659

Rounded

17.0 · 17.03 · 17.029

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.029386

Prime factorization

2 × 5 × 29

Cube root

6.619106

How to simplify √290

The prime factorization of 290 is 2 × 5 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √290 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 290, 2, 5 and 29 appear an odd number of times, so √290 is irrational and 17.0293863659 is a rounded value.

Where √290 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √290 lies between 17 and 18. 290 is 1 above 289 and 34 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.0286 (0% low)
• Tangent from 17, i.e. 17 + 1 ÷ 34: 17.0294 (0% high)
• Tangent from 18, i.e. 18 − 34 ÷ 36: 17.0556 (0.15% high)

For √290 the tangent at 17 wins, missing by only 0. Tangent estimates shine when the number sits close to a perfect square — here 290 is just 1 above 289.

Finding √290 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

Because the starting guess was already close, two steps are enough to match √290 = 17.0293863659 to every decimal shown.

√290 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √290 the pattern is [17; 34] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 290 is one more than a perfect square (17² + 1). A pattern that never ends is one more proof that √290 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 − 290y² = 1. Its smallest solution in positive whole numbers is x = 579, y = 34. Because the period is odd, the equation with −1 on the right also has a solution: 17² − 290 × 1² = −1.

√290 in geometry and everyday measurements

• A square patio or deck of 290 square feet is about 17.03 ft (17 ft) on each side, so edging all the way around takes 4 × √290 ≈ 68.1 ft.
• 290 = 1² + 17² = 11² + 13², so by the Pythagorean theorem √290 is the diagonal of rectangles measuring 1 × 17 and 11 × 13 — and the distance between the points (0, 0) and (1, 17) on a grid.

Square roots near √290 and related roots

• The cube root of 290 is about 6.619106.
• Squaring undoes the root: (√290)² = 290, while 290² = 84,100 — the number whose square root is 290.

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