Square Root of 136

√136 at a glance

Exact value

2√34

Decimal (10 places)

11.6619037897

Rounded

11.7 · 11.66 · 11.662

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.661904

Prime factorization

2³ × 17

Cube root

5.142563

How to simplify √136

Look for the largest perfect square that divides 136. Here it is 4 (2²), because 136 = 4 × 34 and 34 has no square factor left:

The prime factorization tells the same story: 136 = 2³ × 17. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 × 17 stays inside.

Check: (2√34)² = 2² × 34 = 4 × 34 = 136. As a decimal, 2√34 = 2 × 5.8309518949 ≈ 11.6619037897.

Where √136 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √136 lies between 11 and 12. 136 is 15 above 121 and 8 below 144, so the root is closer to 12.

• Straight line between 121 and 144: 11.6522 (0.08% low)
• Tangent from 11, i.e. 11 + 15 ÷ 22: 11.6818 (0.17% high)
• Tangent from 12, i.e. 12 − 8 ÷ 24: 11.6667 (0.04% high)

For √136 the tangent at 12 wins, missing by only 0.0048. Tangent estimates shine when the number sits close to a perfect square — here 136 is just 8 below 144.

Finding √136 with the Babylonian method

Picture a rectangle with an area of 136 and one side x; the other side must be 136 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √136.

Start from the nearest whole number, 12 (12² = 144):

The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √136 = 11.6619037897 to every decimal shown.

√136 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √136 the pattern is [11; 1, 1, 1, 22] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √136 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 − 136y² = 1. Its smallest solution in positive whole numbers is x = 35, y = 3.

√136 in geometry and everyday measurements

• A square room or garden bed covering 136 square feet measures about 11.66 ft (11 ft 8 in) along each wall.
• 136 = 6² + 10², so by the Pythagorean theorem √136 is the diagonal of a 6 × 10 rectangle — and the distance between the points (0, 0) and (6, 10) on a grid.
• Since √136 = 2√34, a length of √136 is exactly 2 copies of the length √34 laid end to end.

Square roots near √136 and related roots

• The cube root of 136 is about 5.142563.
• Four times the radicand doubles the root: √544 = 2 × √136 ≈ 23.323808.

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