√133 at a glance
- Exact value
- √133
- Decimal (10 places)
- 11.5325625947
- Rounded
- 11.5 · 11.53 · 11.533
- Perfect square?
- No — between 11² and 12²
- Rational?
- Irrational
- Both square roots
- ±11.532563
- Prime factorization
- 7 × 19
- Cube root
- 5.104469
How to simplify √133
The prime factorization of 133 is 7 × 19. Every prime appears only once, so there is no pair to bring outside the radical — √133 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 133, 7 and 19 appear an odd number of times, so √133 is irrational and 11.5325625947 is a rounded value.
Where √133 sits between perfect squares
121 = 11² and 144 = 12² are the nearest perfect squares, so √133 lies between 11 and 12. 133 is 12 above 121 and 11 below 144, so the root is closer to 12.
- Straight line between 121 and 144: 11.5217 (0.09% low)
- Tangent from 11, i.e. 11 + 12 ÷ 22: 11.5455 (0.11% high)
- Tangent from 12, i.e. 12 − 11 ÷ 24: 11.5417 (0.08% high)
For √133 the tangent at 12 wins, missing by only 0.0091. Tangent estimates shine when the number sits close to a perfect square — here 133 is just 11 below 144.
Finding √133 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 133: following the tangent line down to zero simplifies to averaging x with 133 ÷ x.
Start from the nearest whole number, 12 (12² = 144):
| Step | Guess x | 133 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 12.0000000000 | 11.0833333333 | 11.5416666667 | 2 |
| 2 | 11.5416666667 | 11.5234657040 | 11.5325661853 | 5 |
| 3 | 11.5325661853 | 11.5325590040 | 11.5325625947 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √133 = 11.5325625947 to every decimal shown.
√133 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √133 the pattern is [11; 1, 1, 7, 5, 1, 1, 1, 2, 1, 1, 1, 5, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √133 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 |
|---|---|---|
| 11/1 | 11.0000000000 | 5.3 × 10⁻¹ |
| 12/1 | 12.0000000000 | 4.7 × 10⁻¹ |
| 23/2 | 11.5000000000 | 3.3 × 10⁻² |
| 173/15 | 11.5333333333 | 7.7 × 10⁻⁴ |
| 888/77 | 11.5324675325 | 9.5 × 10⁻⁵ |
| 1,061/92 | 11.5326086957 | 4.6 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 133y² = 1. Its smallest solution in positive whole numbers is x = 2,588,599, y = 224,460.
√133 in geometry and everyday measurements
- A square room or garden bed covering 133 square feet measures about 11.53 ft (11 ft 6 in) along each wall.
- 133 is not a sum of two whole-number squares — the prime factor 7 and 19 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √133 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 6 × 9 box, because 4² + 6² + 9² = 133.
Square roots near √133 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √130 | √130 | 11.4018 | No |
| √131 | √131 | 11.4455 | No |
| √132 | 2√33 | 11.4891 | No |
| √133 | √133 | 11.5326 | No |
| √134 | √134 | 11.5758 | No |
| √135 | 3√15 | 11.6190 | No |
| √136 | 2√34 | 11.6619 | No |
- The cube root of 133 is about 5.104469.
- Four times the radicand doubles the root: √532 = 2 × √133 ≈ 23.065125.
Frequently asked questions
What is the square root of 133?
The square root of 133 is √133, about 11.5325625947. The negative root, −11.532563, also squares to 133.
Is the square root of 133 rational or irrational?
Irrational. 133 is not a perfect square — it falls between 121 and 144 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √133 be simplified?
No. 133 = 7 × 19 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √133 rounded to two decimal places?
√133 ≈ 11.53 to two decimal places (11.5 to one, 11.533 to three). Check: 11.53² = 132.9409, close to 133.