√177 at a glance
- Exact value
- √177
- Decimal (10 places)
- 13.3041346957
- Rounded
- 13.3 · 13.30 · 13.304
- Perfect square?
- No — between 13² and 14²
- Rational?
- Irrational
- Both square roots
- ±13.304135
- Prime factorization
- 3 × 59
- Cube root
- 5.614672
How to simplify √177
The prime factorization of 177 is 3 × 59. Every prime appears only once, so there is no pair to bring outside the radical — √177 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 177, 3 and 59 appear an odd number of times, so √177 is irrational and 13.3041346957 is a rounded value.
Where √177 sits between perfect squares
169 = 13² and 196 = 14² are the nearest perfect squares, so √177 lies between 13 and 14. 177 is 8 above 169 and 19 below 196, so the root is closer to 13.
- Straight line between 169 and 196: 13.2963 (0.06% low)
- Tangent from 13, i.e. 13 + 8 ÷ 26: 13.3077 (0.03% high)
- Tangent from 14, i.e. 14 − 19 ÷ 28: 13.3214 (0.13% high)
For √177 the tangent at 13 wins, missing by only 0.0036. Tangent estimates shine when the number sits close to a perfect square — here 177 is just 8 above 169.
Finding √177 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 177: following the tangent line down to zero simplifies to averaging x with 177 ÷ x.
Start from the nearest whole number, 13 (13² = 169):
| Step | Guess x | 177 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 13.0000000000 | 13.6153846154 | 13.3076923077 | 2 |
| 2 | 13.3076923077 | 13.3005780347 | 13.3041351712 | 6 |
| 3 | 13.3041351712 | 13.3041342201 | 13.3041346957 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √177 = 13.3041346957 to every decimal shown.
√177 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √177 the pattern is [13; 3, 3, 2, 8, 2, 3, 3, 26] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √177 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 | 3.0 × 10⁻¹ |
| 40/3 | 13.3333333333 | 2.9 × 10⁻² |
| 133/10 | 13.3000000000 | 4.1 × 10⁻³ |
| 306/23 | 13.3043478261 | 2.1 × 10⁻⁴ |
| 2,581/194 | 13.3041237113 | 1.1 × 10⁻⁵ |
| 5,468/411 | 13.3041362530 | 1.6 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 177y² = 1. Its smallest solution in positive whole numbers is x = 62,423, y = 4,692.
√177 in geometry and everyday measurements
- A square room or garden bed covering 177 square feet measures about 13.3 ft (13 ft 4 in) along each wall.
- 177 is not a sum of two whole-number squares — the prime factor 3 and 59 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √177 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 2 × 13 box, because 2² + 2² + 13² = 177.
Square roots near √177 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √174 | √174 | 13.1909 | No |
| √175 | 5√7 | 13.2288 | No |
| √176 | 4√11 | 13.2665 | No |
| √177 | √177 | 13.3041 | No |
| √178 | √178 | 13.3417 | No |
| √179 | √179 | 13.3791 | No |
| √180 | 6√5 | 13.4164 | No |
- The cube root of 177 is about 5.614672.
- Four times the radicand doubles the root: √708 = 2 × √177 ≈ 26.608269.
Frequently asked questions
What is the square root of 177?
The square root of 177 is √177, about 13.3041346957. The negative root, −13.304135, also squares to 177.
Is the square root of 177 rational or irrational?
Irrational. 177 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 √177 be simplified?
No. 177 = 3 × 59 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √177 rounded to two decimal places?
√177 ≈ 13.30 to two decimal places (13.3 to one, 13.304 to three). Check: 13.30² = 176.89, close to 177.