Square Root of 171

√171 at a glance

Exact value

3√19

Decimal (10 places)

13.0766968306

Rounded

13.1 · 13.08 · 13.077

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.076697

Prime factorization

3² × 19

Cube root

5.550499

How to simplify √171

Look for the largest perfect square that divides 171. Here it is 9 (3²), because 171 = 9 × 19 and 19 has no square factor left:

The prime factorization tells the same story: 171 = 3² × 19. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 19 stays inside.

Check: (3√19)² = 3² × 19 = 9 × 19 = 171. As a decimal, 3√19 = 3 × 4.3588989435 ≈ 13.0766968306.

Where √171 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √171 lies between 13 and 14. 171 is 2 above 169 and 25 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.0741 (0.02% low)
• Tangent from 13, i.e. 13 + 2 ÷ 26: 13.0769 (0% high)
• Tangent from 14, i.e. 14 − 25 ÷ 28: 13.1071 (0.23% high)

For √171 the tangent at 13 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 171 is just 2 above 169.

Finding √171 with the Babylonian method

If a guess is too big, 171 ÷ 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√171) in one step.

Start from the nearest whole number, 13 (13² = 169):

The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √171 = 13.0766968306 to every decimal shown.

√171 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √171 the pattern is [13; 13, 26] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √171 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 − 171y² = 1. Its smallest solution in positive whole numbers is x = 170, y = 13.

√171 in geometry and everyday measurements

• A square room or garden bed covering 171 square feet measures about 13.08 ft (13 ft 1 in) along each wall.
• 171 is not a sum of two whole-number squares — the prime factor 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √171 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 1 × 13 box, because 1² + 1² + 13² = 171.
• Since √171 = 3√19, a length of √171 is exactly 3 copies of the length √19 laid end to end.

Square roots near √171 and related roots

• The cube root of 171 is about 5.550499.
• Four times the radicand doubles the root: √684 = 2 × √171 ≈ 26.153394.

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