Square Root of 308

√308 at a glance

Exact value

2√77

Decimal (10 places)

17.5499287748

Rounded

17.5 · 17.55 · 17.550

Perfect square?

No — between 17² and 18²

Rational?

Irrational

Both square roots

±17.549929

Prime factorization

2² × 7 × 11

Cube root

6.753313

How to simplify √308

Look for the largest perfect square that divides 308. Here it is 4 (2²), because 308 = 4 × 77 and 77 has no square factor left:

The prime factorization tells the same story: 308 = 2² × 7 × 11. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 × 11 stays inside.

Check: (2√77)² = 2² × 77 = 4 × 77 = 308. As a decimal, 2√77 = 2 × 8.7749643874 ≈ 17.5499287748.

Where √308 sits between perfect squares

289 = 17² and 324 = 18² are the nearest perfect squares, so √308 lies between 17 and 18. 308 is 19 above 289 and 16 below 324, so the root is closer to 18.

• Straight line between 289 and 324: 17.5429 (0.04% low)
• Tangent from 17, i.e. 17 + 19 ÷ 34: 17.5588 (0.05% high)
• Tangent from 18, i.e. 18 − 16 ÷ 36: 17.5556 (0.03% high)

For √308 the tangent at 18 wins, missing by only 0.0056. Tangent estimates shine when the number sits close to a perfect square — here 308 is just 16 below 324.

Finding √308 with the Babylonian method

Picture a rectangle with an area of 308 and one side x; the other side must be 308 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √308.

Start from the nearest whole number, 18 (18² = 324):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √308 = 17.5499287748 to every decimal shown.

√308 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √308 the pattern is [17; 1, 1, 4, 1, 1, 34] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √308 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 − 308y² = 1. Its smallest solution in positive whole numbers is x = 351, y = 20.

√308 in geometry and everyday measurements

• A square patio or deck of 308 square feet is about 17.55 ft (17 ft 7 in) on each side, so edging all the way around takes 4 × √308 ≈ 70.2 ft.
• 308 is not a sum of two whole-number squares — the prime factor 7 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √308 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 6 × 16 box, because 4² + 6² + 16² = 308.
• Since √308 = 2√77, a length of √308 is exactly 2 copies of the length √77 laid end to end.

Square roots near √308 and related roots

• The cube root of 308 is about 6.753313.
• Because 308 = 4 × 77, the root is twice √77: 2 × 8.774964 ≈ 17.549929.

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