Square Root of 170

√170 at a glance

Exact value

√170

Decimal (10 places)

13.0384048104

Rounded

13.0 · 13.04 · 13.038

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.038405

Prime factorization

2 × 5 × 17

Cube root

5.539658

How to simplify √170

The prime factorization of 170 is 2 × 5 × 17. Every prime appears only once, so there is no pair to bring outside the radical — √170 is already in simplest form (a "square-free" radicand).

A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 170, 2, 5 and 17 appear an odd number of times, so √170 is irrational and 13.0384048104 is a rounded value.

Where √170 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √170 lies between 13 and 14. 170 is 1 above 169 and 26 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.0370 (0.01% low)
• Tangent from 13, i.e. 13 + 1 ÷ 26: 13.0385 (0% high)
• Tangent from 14, i.e. 14 − 26 ÷ 28: 13.0714 (0.25% high)

For √170 the tangent at 13 wins, missing by only 0.0001. Tangent estimates shine when the number sits close to a perfect square — here 170 is just 1 above 169.

Finding √170 with the Babylonian method

The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.

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

The count of correct decimals went 4, 9 and all 10 over 3 steps — roughly doubling each time — until the guess matched √170 = 13.0384048104 to every decimal shown.

√170 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √170 the pattern is [13; 26] with the block of 1 term after the semicolon repeating forever — the simplest possible pattern, which happens exactly when 170 is one more than a perfect square (13² + 1). A pattern that never ends is one more proof that √170 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 − 170y² = 1. Its smallest solution in positive whole numbers is x = 339, y = 26. Because the period is odd, the equation with −1 on the right also has a solution: 13² − 170 × 1² = −1.

√170 in geometry and everyday measurements

• A square room or garden bed covering 170 square feet measures about 13.04 ft (13 ft) along each wall.
• 170 = 1² + 13² = 7² + 11², so by the Pythagorean theorem √170 is the diagonal of rectangles measuring 1 × 13 and 7 × 11 — and the distance between the points (0, 0) and (1, 13) on a grid.

Square roots near √170 and related roots

• The cube root of 170 is about 5.539658.
• Four times the radicand doubles the root: √680 = 2 × √170 ≈ 26.07681.

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