√33 at a glance
- Exact value
- √33
- Decimal (10 places)
- 5.7445626465
- Rounded
- 5.7 · 5.74 · 5.745
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.744563
- Prime factorization
- 3 × 11
- Cube root
- 3.207534
How to simplify √33
The prime factorization of 33 is 3 × 11. Every prime appears only once, so there is no pair to bring outside the radical — √33 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 33, 3 and 11 appear an odd number of times, so √33 is irrational and 5.7445626465 is a rounded value.
Where √33 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √33 lies between 5 and 6. 33 is 8 above 25 and 3 below 36, so the root is closer to 6.
- Straight line between 25 and 36: 5.7273 (0.3% low)
- Tangent from 5, i.e. 5 + 8 ÷ 10: 5.8000 (0.97% high)
- Tangent from 6, i.e. 6 − 3 ÷ 12: 5.7500 (0.09% high)
For √33 the tangent at 6 wins, missing by only 0.0054. Tangent estimates shine when the number sits close to a perfect square — here 33 is just 3 below 36.
Finding √33 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 33: following the tangent line down to zero simplifies to averaging x with 33 ÷ x.
Start from the nearest whole number, 6 (6² = 36):
| Step | Guess x | 33 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 6.0000000000 | 5.5000000000 | 5.7500000000 | 2 |
| 2 | 5.7500000000 | 5.7391304348 | 5.7445652174 | 5 |
| 3 | 5.7445652174 | 5.7445600757 | 5.7445626465 | 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 √33 = 5.7445626465 to every decimal shown.
√33 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √33 the pattern is [5; 1, 2, 1, 10] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √33 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 7.4 × 10⁻¹ |
| 6/1 | 6.0000000000 | 2.6 × 10⁻¹ |
| 17/3 | 5.6666666667 | 7.8 × 10⁻² |
| 23/4 | 5.7500000000 | 5.4 × 10⁻³ |
| 247/43 | 5.7441860465 | 3.8 × 10⁻⁴ |
| 270/47 | 5.7446808511 | 1.2 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 33y² = 1. Its smallest solution in positive whole numbers is x = 23, y = 4.
√33 in geometry and everyday measurements
- A square room or garden bed covering 33 square feet measures about 5.74 ft (5 ft 9 in) along each wall.
- 33 is not a sum of two whole-number squares — the prime factor 3 and 11 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √33 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 4 box, because 1² + 4² + 4² = 33.
Square roots near √33 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √30 | √30 | 5.4772 | No |
| √31 | √31 | 5.5678 | No |
| √32 | 4√2 | 5.6569 | No |
| √33 | √33 | 5.7446 | No |
| √34 | √34 | 5.8310 | No |
| √35 | √35 | 5.9161 | No |
| √36 | 6 | 6.0000 | Yes |
- The cube root of 33 is about 3.207534.
- Four times the radicand doubles the root: √132 = 2 × √33 ≈ 11.489125.
Frequently asked questions
What is the square root of 33?
The square root of 33 is √33, about 5.7445626465. The negative root, −5.744563, also squares to 33.
Is the square root of 33 rational or irrational?
Irrational. 33 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √33 be simplified?
No. 33 = 3 × 11 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √33 rounded to two decimal places?
√33 ≈ 5.74 to two decimal places (5.7 to one, 5.745 to three). Check: 5.74² = 32.9476, close to 33.