Square Root of 68

√68 at a glance

Exact value

2√17

Decimal (10 places)

8.2462112512

Rounded

8.2 · 8.25 · 8.246

Perfect square?

No — between 8² and 9²

Rational?

Irrational

Both square roots

±8.246211

Prime factorization

2² × 17

Cube root

4.081655

How to simplify √68

Look for the largest perfect square that divides 68. Here it is 4 (2²), because 68 = 4 × 17 and 17 has no square factor left:

The prime factorization tells the same story: 68 = 2² × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 17 stays inside.

Check: (2√17)² = 2² × 17 = 4 × 17 = 68. As a decimal, 2√17 = 2 × 4.1231056256 ≈ 8.2462112512.

Where √68 sits between perfect squares

64 = 8² and 81 = 9² are the nearest perfect squares, so √68 lies between 8 and 9. 68 is 4 above 64 and 13 below 81, so the root is closer to 8.

• Straight line between 64 and 81: 8.2353 (0.13% low)
• Tangent from 8, i.e. 8 + 4 ÷ 16: 8.2500 (0.05% high)
• Tangent from 9, i.e. 9 − 13 ÷ 18: 8.2778 (0.38% high)

For √68 the tangent at 8 wins, missing by only 0.0038. Tangent estimates shine when the number sits close to a perfect square — here 68 is just 4 above 64.

Finding √68 with the Babylonian method

Picture a rectangle with an area of 68 and one side x; the other side must be 68 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √68.

Start from the nearest whole number, 8 (8² = 64):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √68 = 8.2462112512 to every decimal shown.

√68 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √68 the pattern is [8; 4, 16] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √68 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 − 68y² = 1. Its smallest solution in positive whole numbers is x = 33, y = 4.

√68 in geometry and everyday measurements

• A square room or garden bed covering 68 square feet measures about 8.25 ft (8 ft 3 in) along each wall.
• 68 = 2² + 8², so by the Pythagorean theorem √68 is the diagonal of a 2 × 8 rectangle — and the distance between the points (0, 0) and (2, 8) on a grid.
• Since √68 = 2√17, a length of √68 is exactly 2 copies of the length √17 laid end to end.

Square roots near √68 and related roots

• The cube root of 68 is about 4.081655.
• Four times the radicand doubles the root: √272 = 2 × √68 ≈ 16.492423.

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