Square Root of 178

√178 at a glance

Exact value

√178

Decimal (10 places)

13.3416640641

Rounded

13.3 · 13.34 · 13.342

Perfect square?

No — between 13² and 14²

Rational?

Irrational

Both square roots

±13.341664

Prime factorization

2 × 89

Cube root

5.625226

How to simplify √178

The prime factorization of 178 is 2 × 89. Every prime appears only once, so there is no pair to bring outside the radical — √178 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 178, 2 and 89 appear an odd number of times, so √178 is irrational and 13.3416640641 is a rounded value.

Where √178 sits between perfect squares

169 = 13² and 196 = 14² are the nearest perfect squares, so √178 lies between 13 and 14. 178 is 9 above 169 and 18 below 196, so the root is closer to 13.

• Straight line between 169 and 196: 13.3333 (0.06% low)
• Tangent from 13, i.e. 13 + 9 ÷ 26: 13.3462 (0.03% high)
• Tangent from 14, i.e. 14 − 18 ÷ 28: 13.3571 (0.12% high)

For √178 the tangent at 13 wins, missing by only 0.0045. Tangent estimates shine when the number sits close to a perfect square — here 178 is just 9 above 169.

Finding √178 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 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √178 = 13.3416640641 to every decimal shown.

√178 as a continued fraction

Every irrational square root has a continued fraction that repeats. For √178 the pattern is [13; 2, 1, 12, 1, 2, 26] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √178 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 − 178y² = 1. Its smallest solution in positive whole numbers is x = 1,601, y = 120.

√178 in geometry and everyday measurements

• A square room or garden bed covering 178 square feet measures about 13.34 ft (13 ft 4 in) along each wall.
• 178 = 3² + 13², so by the Pythagorean theorem √178 is the diagonal of a 3 × 13 rectangle — and the distance between the points (0, 0) and (3, 13) on a grid.

Square roots near √178 and related roots

• The cube root of 178 is about 5.625226.
• Four times the radicand doubles the root: √712 = 2 × √178 ≈ 26.683328.

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