Square Root of 305

√305 at a glance

Exact value

√305

Decimal (10 places)

17.4642491966

Rounded

17.5 · 17.46 · 17.464

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.464249

Prime factorization

5 × 61

Cube root

6.731315

How to simplify √305

The prime factorization of 305 is 5 × 61. Every prime appears only once, so there is no pair to bring outside the radical — √305 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 305, 5 and 61 appear an odd number of times, so √305 is irrational and 17.4642491966 is a rounded value.

Where √305 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √305 lies between 17 and 18. 305 is 16 above 289 and 19 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.4571 (0.04% low)
• Tangent from 17, i.e. 17 + 16 ÷ 34: 17.4706 (0.04% high)
• Tangent from 18, i.e. 18 − 19 ÷ 36: 17.4722 (0.05% high)

For √305 the tangent at 17 wins, missing by only 0.0063. Tangent estimates shine when the number sits close to a perfect square — here 305 is just 16 above 289.

Finding √305 with the Babylonian method

This is Newton’s method applied to f(x) = x² − 305: following the tangent line down to zero simplifies to averaging x with 305 ÷ x.

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

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √305 = 17.4642491966 to every decimal shown.

√305 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √305 the pattern is [17; 2, 6, 2, 34] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √305 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 − 305y² = 1. Its smallest solution in positive whole numbers is x = 489, y = 28.

√305 in geometry and everyday measurements

• A square patio or deck of 305 square feet is about 17.46 ft (17 ft 6 in) on each side, so edging all the way around takes 4 × √305 ≈ 69.9 ft.
• 305 = 4² + 17² = 7² + 16², so by the Pythagorean theorem √305 is the diagonal of rectangles measuring 4 × 17 and 7 × 16 — and the distance between the points (0, 0) and (4, 17) on a grid.

Square roots near √305 and related roots

• The cube root of 305 is about 6.731315.
• Squaring undoes the root: (√305)² = 305, while 305² = 93,025 — the number whose square root is 305.

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