Square Root of 15
The square root of 15 is approximately 3.872983, or exactly √15 in simplified radical form. No, 15 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √15 sits
3² = 9 and 4² = 16, and 15 falls between them, so √15 is between 3 and 4. It's closer to 4 (6 above 9, 1 below 16), which matches the decimal 3.872983.
| k | k² | Compared to 15 |
|---|---|---|
| 1 | 1 | 14 below |
| 2 | 4 | 11 below |
| 3 | 9 | 6 below |
| 4 | 16 | 1 above |
| 5 | 25 | 10 above |
| 6 | 36 | 21 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 3), divide 15 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 15 ÷ x | New guess (x + 15÷x) ÷ 2 |
|---|---|---|---|
| 1 | 3 | 5 | 4 |
| 2 | 4 | 3.75 | 3.875 |
| 3 | 3.875 | 3.870968 | 3.872984 |
| 4 | 3.872984 | 3.872983 | 3.872983 |
After 4 passes the guess is 3.872983, already matching √15 = 3.872983 to six decimal places.
Exact form from the prime factors
15 = 3 × 5. No prime appears twice, so nothing can be pulled out of the root -- √15 is already in simplest form.
Frequently Asked Questions
What is the square root of 15?
The square root of 15 is approximately 3.872983, or exactly √15 in simplified radical form.
Is 15 a perfect square?
No, 15 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √15 between?
Between 3 and 4, because 3² = 9 and 4² = 16.
What are the square roots of the numbers right next to 15?
√14 is about 3.7417, and √16 is exactly 4. 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 √15 squared?
Exactly 15 -- squaring undoes the square root. Squaring the rounded decimal 3.872983 gives 15, which is why the radical form √15 is the exact answer and the decimal is an approximation.
What is the negative square root of 15?
−3.872983. Every positive number has two square roots, one positive and one negative, because (−3.872983)² is also 15. The radical sign √ by convention means the positive one.
More Number Theory
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