Square Root of 34
The square root of 34 is approximately 5.830952, or exactly √34 in simplified radical form. No, 34 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √34 sits
5² = 25 and 6² = 36, and 34 falls between them, so √34 is between 5 and 6. It's closer to 6 (9 above 25, 2 below 36), which matches the decimal 5.830952.
| k | k² | Compared to 34 |
|---|---|---|
| 3 | 9 | 25 below |
| 4 | 16 | 18 below |
| 5 | 25 | 9 below |
| 6 | 36 | 2 above |
| 7 | 49 | 15 above |
| 8 | 64 | 30 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 34 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 34 ÷ x | New guess (x + 34÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 6.8 | 5.9 |
| 2 | 5.9 | 5.762712 | 5.831356 |
| 3 | 5.831356 | 5.830548 | 5.830952 |
| 4 | 5.830952 | 5.830952 | 5.830952 |
After 4 passes the guess is 5.830952, already matching √34 = 5.830952 to six decimal places.
Exact form from the prime factors
34 = 2 × 17. No prime appears twice, so nothing can be pulled out of the root -- √34 is already in simplest form.
Frequently Asked Questions
What is the square root of 34?
The square root of 34 is approximately 5.830952, or exactly √34 in simplified radical form.
Is 34 a perfect square?
No, 34 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √34 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 34?
√33 is about 5.7446, and √35 is about 5.9161. 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 √34 squared?
Exactly 34 -- squaring undoes the square root. Squaring the rounded decimal 5.830952 gives 34, which is why the radical form √34 is the exact answer and the decimal is an approximation.
What is the negative square root of 34?
−5.830952. Every positive number has two square roots, one positive and one negative, because (−5.830952)² is also 34. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 34
- GCF Calculator
- LCM Calculator