√174 at a glance
- Exact value
- √174
- Decimal (10 places)
- 13.1909059583
- Rounded
- 13.2 · 13.19 · 13.191
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.190906
- Prime factorization
- 2 × 3 × 29
- Cube root
- 5.582770
How to simplify √174
The prime factorization of 174 is 2 × 3 × 29. Every prime appears only once, so there is no pair to bring outside the radical — √174 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 174, 2, 3 and 29 appear an odd number of times, so √174 is irrational and 13.1909059583 is a rounded value.
Where √174 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √174 lies between 13 and 14. 174 is 5 above 169 and 22 below 196, so the root is closer to 13.
- Straight line between 169 and 196: 13.1852 (0.04% low)
- Tangent from 13, i.e. 13 + 5 ÷ 26: 13.1923 (0.01% high)
- Tangent from 14, i.e. 14 − 22 ÷ 28: 13.2143 (0.18% high)
For √174 the tangent at 13 wins, missing by only 0.0014. Tangent estimates shine when the number sits close to a perfect square — here 174 is just 5 above 169.
Finding √174 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):
| Step | Guess x | 174 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 13.3846153846 | 13.1923076923 | 2 |
| 2 | 13.1923076923 | 13.1895043732 | 13.1909060327 | 7 |
| 3 | 13.1909060327 | 13.1909058838 | 13.1909059583 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √174 = 13.1909059583 to every decimal shown.
√174 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √174 the pattern is [13; 5, 4, 5, 26] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √174 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 13/1 | 13.0000000000 | 1.9 × 10⁻¹ |
| 66/5 | 13.2000000000 | 9.1 × 10⁻³ |
| 277/21 | 13.1904761905 | 4.3 × 10⁻⁴ |
| 1,451/110 | 13.1909090909 | 3.1 × 10⁻⁶ |
| 38,003/2,881 | 13.1909059354 | 2.3 × 10⁻⁸ |
| 191,466/14,515 | 13.1909059594 | 1.1 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 174y² = 1. Its smallest solution in positive whole numbers is x = 1,451, y = 110.
√174 in geometry and everyday measurements
- A square room or garden bed covering 174 square feet measures about 13.19 ft (13 ft 2 in) along each wall.
- 174 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 √174 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 13 box, because 1² + 2² + 13² = 174.
Square roots near √174 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √171 | 3√19 | 13.0767 | No |
| √172 | 2√43 | 13.1149 | No |
| √173 | √173 | 13.1529 | No |
| √174 | √174 | 13.1909 | No |
| √175 | 5√7 | 13.2288 | No |
| √176 | 4√11 | 13.2665 | No |
| √177 | √177 | 13.3041 | No |
- The cube root of 174 is about 5.582770.
- Four times the radicand doubles the root: √696 = 2 × √174 ≈ 26.381812.
Frequently asked questions
What is the square root of 174?
The square root of 174 is √174, about 13.1909059583. The negative root, −13.190906, also squares to 174.
Is the square root of 174 rational or irrational?
Irrational. 174 is not a perfect square — it falls between 169 and 196 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √174 be simplified?
No. 174 = 2 × 3 × 29 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √174 rounded to two decimal places?
√174 ≈ 13.19 to two decimal places (13.2 to one, 13.191 to three). Check: 13.19² = 173.9761, close to 174.