Square Root of 35
The square root of 35 is approximately 5.91608, or exactly √35 in simplified radical form. No, 35 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √35 sits
5² = 25 and 6² = 36, and 35 falls between them, so √35 is between 5 and 6. It's closer to 6 (10 above 25, 1 below 36), which matches the decimal 5.91608.
| k | k² | Compared to 35 |
|---|---|---|
| 3 | 9 | 26 below |
| 4 | 16 | 19 below |
| 5 | 25 | 10 below |
| 6 | 36 | 1 above |
| 7 | 49 | 14 above |
| 8 | 64 | 29 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 35 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 35 ÷ x | New guess (x + 35÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 7 | 6 |
| 2 | 6 | 5.833333 | 5.916667 |
| 3 | 5.916667 | 5.915493 | 5.91608 |
| 4 | 5.91608 | 5.91608 | 5.91608 |
After 4 passes the guess is 5.91608, already matching √35 = 5.91608 to six decimal places.
Exact form from the prime factors
35 = 5 × 7. No prime appears twice, so nothing can be pulled out of the root -- √35 is already in simplest form.
Frequently Asked Questions
What is the square root of 35?
The square root of 35 is approximately 5.91608, or exactly √35 in simplified radical form.
Is 35 a perfect square?
No, 35 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √35 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 35?
√34 is about 5.831, and √36 is exactly 6. 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 √35 squared?
Exactly 35 -- squaring undoes the square root. Squaring the rounded decimal 5.91608 gives 35, which is why the radical form √35 is the exact answer and the decimal is an approximation.
What is the negative square root of 35?
−5.91608. Every positive number has two square roots, one positive and one negative, because (−5.91608)² is also 35. The radical sign √ by convention means the positive one.
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