Square Root of 135

√135 at a glance

Exact value

3√15

Decimal (10 places)

11.6189500386

Rounded

11.6 · 11.62 · 11.619

Perfect square?

No — between 11² and 12²

Rational?

Irrational

Both square roots

±11.618950

Prime factorization

3³ × 5

Cube root

5.129928

How to simplify √135

Look for the largest perfect square that divides 135. Here it is 9 (3²), because 135 = 9 × 15 and 15 has no square factor left:

The prime factorization tells the same story: 135 = 3³ × 5. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 3 × 5 stays inside.

Check: (3√15)² = 3² × 15 = 9 × 15 = 135. As a decimal, 3√15 = 3 × 3.8729833462 ≈ 11.6189500386.

Where √135 sits between perfect squares

121 = 11² and 144 = 12² are the nearest perfect squares, so √135 lies between 11 and 12. 135 is 14 above 121 and 9 below 144, so the root is closer to 12.

• Straight line between 121 and 144: 11.6087 (0.09% low)
• Tangent from 11, i.e. 11 + 14 ÷ 22: 11.6364 (0.15% high)
• Tangent from 12, i.e. 12 − 9 ÷ 24: 11.6250 (0.05% high)

For √135 the tangent at 12 wins, missing by only 0.006. Tangent estimates shine when the number sits close to a perfect square — here 135 is just 9 below 144.

Finding √135 with the Babylonian method

If a guess is too big, 135 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√135) in one step.

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

The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √135 = 11.6189500386 to every decimal shown.

√135 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √135 the pattern is [11; 1, 1, 1, 1, 1, 1, 1, 22] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √135 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 − 135y² = 1. Its smallest solution in positive whole numbers is x = 244, y = 21.

√135 in geometry and everyday measurements

• A square room or garden bed covering 135 square feet measures about 11.62 ft (11 ft 7 in) along each wall.
• 135 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √135 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √135 as its space diagonal.
• Since √135 = 3√15, a length of √135 is exactly 3 copies of the length √15 laid end to end.

Square roots near √135 and related roots

• The cube root of 135 is about 5.129928.
• Four times the radicand doubles the root: √540 = 2 × √135 ≈ 23.2379.

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