√167 at a glance
- Exact value
- √167
- Decimal (10 places)
- 12.9228479833
- Rounded
- 12.9 · 12.92 · 12.923
- Perfect square?
- No — between 12² and 13²
- Rational?
- Irrational
- Both square roots
- ±12.922848
- Prime factorization
- 167
- Cube root
- 5.506878
How to simplify √167
167 is a prime number, so its only factors are 1 and 167. There is no perfect-square factor to pull out, which means √167 is already in its simplest radical form.
The square root of any prime is irrational. If √167 were a fraction a/b in lowest terms, then a² = 167b², so 167 would divide a — and then 167 would divide b too, contradicting “lowest terms.” That is why the decimal 12.9228479833 is only a rounded value.
Where √167 sits between perfect squares
144 = 12² and 169 = 13² are the nearest perfect squares, so √167 lies between 12 and 13. 167 is 23 above 144 and 2 below 169, so the root is closer to 13.
- Straight line between 144 and 169: 12.9200 (0.02% low)
- Tangent from 12, i.e. 12 + 23 ÷ 24: 12.9583 (0.27% high)
- Tangent from 13, i.e. 13 − 2 ÷ 26: 12.9231 (0% high)
For √167 the tangent at 13 wins, missing by only 0.0002. Tangent estimates shine when the number sits close to a perfect square — here 167 is just 2 below 169.
Finding √167 with the Babylonian method
If a guess is too big, 167 ÷ 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√167) in one step.
Start from the nearest whole number, 13 (13² = 169):
| Step | Guess x | 167 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 12.8461538462 | 12.9230769231 | 3 |
| 2 | 12.9230769231 | 12.9226190476 | 12.9228479853 | 8 |
| 3 | 12.9228479853 | 12.9228479813 | 12.9228479833 | all 10 shown |
The count of correct decimals went 3, 8 and all 10 over 3 steps — roughly doubling each time — until the guess matched √167 = 12.9228479833 to every decimal shown.
√167 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √167 the pattern is [12; 1, 11, 1, 24] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √167 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 |
|---|---|---|
| 12/1 | 12.0000000000 | 9.2 × 10⁻¹ |
| 13/1 | 13.0000000000 | 7.7 × 10⁻² |
| 155/12 | 12.9166666667 | 6.2 × 10⁻³ |
| 168/13 | 12.9230769231 | 2.3 × 10⁻⁴ |
| 4,187/324 | 12.9228395062 | 8.5 × 10⁻⁶ |
| 4,355/337 | 12.9228486647 | 6.8 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 167y² = 1. Its smallest solution in positive whole numbers is x = 168, y = 13.
√167 in geometry and everyday measurements
- A square room or garden bed covering 167 square feet measures about 12.92 ft (12 ft 11 in) along each wall.
- 167 is not a sum of two whole-number squares — 167 is itself a prime that is one less than a multiple of 4, which rules that out — so √167 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 √167 as its space diagonal.
Square roots near √167 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √164 | 2√41 | 12.8062 | No |
| √165 | √165 | 12.8452 | No |
| √166 | √166 | 12.8841 | No |
| √167 | √167 | 12.9228 | No |
| √168 | 2√42 | 12.9615 | No |
| √169 | 13 | 13.0000 | Yes |
| √170 | √170 | 13.0384 | No |
- The cube root of 167 is about 5.506878.
- Four times the radicand doubles the root: √668 = 2 × √167 ≈ 25.845696.
Frequently asked questions
What is the square root of 167?
The square root of 167 is √167, about 12.9228479833. The negative root, −12.922848, also squares to 167.
Is the square root of 167 rational or irrational?
Irrational. 167 is not a perfect square — it falls between 144 and 169 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √167 be simplified?
No. 167 is prime, so there is no perfect square to take out of the radical.
What is √167 rounded to two decimal places?
√167 ≈ 12.92 to two decimal places (12.9 to one, 12.923 to three). Check: 12.92² = 166.9264, close to 167.