Square Root of 306

√306 at a glance

Exact value

3√34

Decimal (10 places)

17.4928556845

Rounded

17.5 · 17.49 · 17.493

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.492856

Prime factorization

2 × 3² × 17

Cube root

6.738664

How to simplify √306

Look for the largest perfect square that divides 306. Here it is 9 (3²), because 306 = 9 × 34 and 34 has no square factor left:

The prime factorization tells the same story: 306 = 2 × 3² × 17. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 17 stays inside.

Check: (3√34)² = 3² × 34 = 9 × 34 = 306. As a decimal, 3√34 = 3 × 5.8309518949 ≈ 17.4928556845.

Where √306 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √306 lies between 17 and 18. 306 is 17 above 289 and 18 below 324, so the root is closer to 17.

• Straight line between 289 and 324: 17.4857 (0.04% low)
• Tangent from 17, i.e. 17 + 17 ÷ 34: 17.5000 (0.04% high)
• Tangent from 18, i.e. 18 − 18 ÷ 36: 17.5000 (0.04% high)

For √306 the straight-line estimate is the closest of the three. The straight line always undershoots and the tangents always overshoot, because the square-root curve bends downward; near the middle of the gap the undershoot is smaller.

Finding √306 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, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √306 = 17.4928556845 to every decimal shown.

√306 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √306 the pattern is [17; 2, 34] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √306 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 − 306y² = 1. Its smallest solution in positive whole numbers is x = 35, y = 2.

√306 in geometry and everyday measurements

• A square patio or deck of 306 square feet is about 17.49 ft (17 ft 6 in) on each side, so edging all the way around takes 4 × √306 ≈ 70 ft.
• 306 = 9² + 15², so by the Pythagorean theorem √306 is the diagonal of a 9 × 15 rectangle — and the distance between the points (0, 0) and (9, 15) on a grid.
• Since √306 = 3√34, a length of √306 is exactly 3 copies of the length √34 laid end to end.

Square roots near √306 and related roots

• The cube root of 306 is about 6.738664.
• Squaring undoes the root: (√306)² = 306, while 306² = 93,636 — the number whose square root is 306.

Need a different value? Use the Square root calculator, or browse the Square roots 1–1,000.