Square Root of 33
The square root of 33 is approximately 5.744563, or exactly √33 in simplified radical form. No, 33 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √33 sits
5² = 25 and 6² = 36, and 33 falls between them, so √33 is between 5 and 6. It's closer to 6 (8 above 25, 3 below 36), which matches the decimal 5.744563.
| k | k² | Compared to 33 |
|---|---|---|
| 3 | 9 | 24 below |
| 4 | 16 | 17 below |
| 5 | 25 | 8 below |
| 6 | 36 | 3 above |
| 7 | 49 | 16 above |
| 8 | 64 | 31 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 33 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 33 ÷ x | New guess (x + 33÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 6.6 | 5.8 |
| 2 | 5.8 | 5.689655 | 5.744828 |
| 3 | 5.744828 | 5.744298 | 5.744563 |
| 4 | 5.744563 | 5.744563 | 5.744563 |
After 4 passes the guess is 5.744563, already matching √33 = 5.744563 to six decimal places.
Exact form from the prime factors
33 = 3 × 11. No prime appears twice, so nothing can be pulled out of the root -- √33 is already in simplest form.
Frequently Asked Questions
What is the square root of 33?
The square root of 33 is approximately 5.744563, or exactly √33 in simplified radical form.
Is 33 a perfect square?
No, 33 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √33 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 33?
√32 is about 5.6569, and √34 is about 5.831. Square roots increase more slowly as the number gets bigger, which is why consecutive whole numbers have roots that get closer together the higher you go.
What is √33 squared?
Exactly 33 -- squaring undoes the square root. Squaring the rounded decimal 5.744563 gives 33, which is why the radical form √33 is the exact answer and the decimal is an approximation.
What is the negative square root of 33?
−5.744563. Every positive number has two square roots, one positive and one negative, because (−5.744563)² is also 33. The radical sign √ by convention means the positive one.
More Number Theory
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