Square Root of 302

√302 at a glance

Exact value

√302

Decimal (10 places)

17.3781471970

Rounded

17.4 · 17.38 · 17.378

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.378147

Prime factorization

2 × 151

Cube root

6.709173

How to simplify √302

The prime factorization of 302 is 2 × 151. Every prime appears only once, so there is no pair to bring outside the radical — √302 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 302, 2 and 151 appear an odd number of times, so √302 is irrational and 17.3781471970 is a rounded value.

Where √302 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √302 lies between 17 and 18. 302 is 13 above 289 and 22 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.3714 (0.04% low)
• Tangent from 17, i.e. 17 + 13 ÷ 34: 17.3824 (0.02% high)
• Tangent from 18, i.e. 18 − 22 ÷ 36: 17.3889 (0.06% high)

For √302 the tangent at 17 wins, missing by only 0.0042. Tangent estimates shine when the number sits close to a perfect square — here 302 is just 13 above 289.

Finding √302 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 √302 = 17.3781471970 to every decimal shown.

√302 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √302 the pattern is [17; 2, 1, 1, 1, 4, 2, 1, 16, 1, 2, 4, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √302 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 − 302y² = 1. Its smallest solution in positive whole numbers is x = 4,276,623, y = 246,092.

√302 in geometry and everyday measurements

• A square patio or deck of 302 square feet is about 17.38 ft (17 ft 5 in) on each side, so edging all the way around takes 4 × √302 ≈ 69.5 ft.
• 302 is not a sum of two whole-number squares — the prime factor 151 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √302 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 17 box, because 2² + 3² + 17² = 302.

Square roots near √302 and related roots

• The cube root of 302 is about 6.709173.
• Squaring undoes the root: (√302)² = 302, while 302² = 91,204 — the number whose square root is 302.

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