Square Root of 298

√298 at a glance

Exact value

√298

Decimal (10 places)

17.2626765016

Rounded

17.3 · 17.26 · 17.263

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.262677

Prime factorization

2 × 149

Cube root

6.679420

How to simplify √298

The prime factorization of 298 is 2 × 149. Every prime appears only once, so there is no pair to bring outside the radical — √298 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 298, 2 and 149 appear an odd number of times, so √298 is irrational and 17.2626765016 is a rounded value.

Where √298 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √298 lies between 17 and 18. 298 is 9 above 289 and 26 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.2571 (0.03% low)
• Tangent from 17, i.e. 17 + 9 ÷ 34: 17.2647 (0.01% high)
• Tangent from 18, i.e. 18 − 26 ÷ 36: 17.2778 (0.09% high)

For √298 the tangent at 17 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 298 is just 9 above 289.

Finding √298 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):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √298 = 17.2626765016 to every decimal shown.

√298 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √298 the pattern is [17; 3, 1, 4, 5, 1, 1, 5, 4, 1, 3, 34] with the block of 11 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √298 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 − 298y² = 1. Its smallest solution in positive whole numbers is x = 335,473,872,499, y = 19,433,479,650. Because the period is odd, the equation with −1 on the right also has a solution: 409,557² − 298 × 23,725² = −1.

√298 in geometry and everyday measurements

• A square patio or deck of 298 square feet is about 17.26 ft (17 ft 3 in) on each side, so edging all the way around takes 4 × √298 ≈ 69.1 ft.
• 298 = 3² + 17², so by the Pythagorean theorem √298 is the diagonal of a 3 × 17 rectangle — and the distance between the points (0, 0) and (3, 17) on a grid.

Square roots near √298 and related roots

• The cube root of 298 is about 6.679420.
• Squaring undoes the root: (√298)² = 298, while 298² = 88,804 — the number whose square root is 298.

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