Square Root of 301

√301 at a glance

Exact value

√301

Decimal (10 places)

17.3493515729

Rounded

17.3 · 17.35 · 17.349

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.349352

Prime factorization

7 × 43

Cube root

6.701759

How to simplify √301

The prime factorization of 301 is 7 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √301 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 301, 7 and 43 appear an odd number of times, so √301 is irrational and 17.3493515729 is a rounded value.

Where √301 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √301 lies between 17 and 18. 301 is 12 above 289 and 23 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.3429 (0.04% low)
• Tangent from 17, i.e. 17 + 12 ÷ 34: 17.3529 (0.02% high)
• Tangent from 18, i.e. 18 − 23 ÷ 36: 17.3611 (0.07% high)

For √301 the tangent at 17 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 301 is just 12 above 289.

Finding √301 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 301: following the tangent line down to zero simplifies to averaging x with 301 ÷ x.

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 √301 = 17.3493515729 to every decimal shown.

√301 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √301 the pattern is [17; 2, 1, 6, 3, 1, 2, 2, 1, 1, 8, 11, 2, …] with the block of 26 terms after the semicolon repeating forever (only the first 12 of the 26 are shown). A pattern that never ends is one more proof that √301 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 − 301y² = 1. Its smallest solution in positive whole numbers is x = 5,883,392,537,695, y = 339,113,108,232 — 13 digits for x, even though 301 is small, which is what makes Pell’s equation famous.

√301 in geometry and everyday measurements

• A square patio or deck of 301 square feet is about 17.35 ft (17 ft 4 in) on each side, so edging all the way around takes 4 × √301 ≈ 69.4 ft.
• 301 is not a sum of two whole-number squares — the prime factor 7 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √301 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 6 × 16 box, because 3² + 6² + 16² = 301.

Square roots near √301 and related roots

• The cube root of 301 is about 6.701759.
• Squaring undoes the root: (√301)² = 301, while 301² = 90,601 — the number whose square root is 301.

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