Square Root of 17
The square root of 17 is approximately 4.123106, or exactly √17 in simplified radical form. No, 17 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √17 sits
4² = 16 and 5² = 25, and 17 falls between them, so √17 is between 4 and 5. It's closer to 4 (1 above 16, 8 below 25), which matches the decimal 4.123106.
| k | k² | Compared to 17 |
|---|---|---|
| 2 | 4 | 13 below |
| 3 | 9 | 8 below |
| 4 | 16 | 1 below |
| 5 | 25 | 8 above |
| 6 | 36 | 19 above |
| 7 | 49 | 32 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 17 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 17 ÷ x | New guess (x + 17÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 4.25 | 4.125 |
| 2 | 4.125 | 4.121212 | 4.123106 |
| 3 | 4.123106 | 4.123105 | 4.123106 |
| 4 | 4.123106 | 4.123106 | 4.123106 |
After 4 passes the guess is 4.123106, already matching √17 = 4.123106 to six decimal places.
Exact form from the prime factors
17 = 17. No prime appears twice, so nothing can be pulled out of the root -- √17 is already in simplest form.
Frequently Asked Questions
What is the square root of 17?
The square root of 17 is approximately 4.123106, or exactly √17 in simplified radical form.
Is 17 a perfect square?
No, 17 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √17 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 17?
√16 is exactly 4, and √18 is about 4.2426. 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 √17 squared?
Exactly 17 -- squaring undoes the square root. Squaring the rounded decimal 4.123106 gives 17, which is why the radical form √17 is the exact answer and the decimal is an approximation.
What is the negative square root of 17?
−4.123106. Every positive number has two square roots, one positive and one negative, because (−4.123106)² is also 17. The radical sign √ by convention means the positive one.
More Number Theory
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