Square Root of 14
The square root of 14 is approximately 3.741657, or exactly √14 in simplified radical form. No, 14 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √14 sits
3² = 9 and 4² = 16, and 14 falls between them, so √14 is between 3 and 4. It's closer to 4 (5 above 9, 2 below 16), which matches the decimal 3.741657.
| k | k² | Compared to 14 |
|---|---|---|
| 1 | 1 | 13 below |
| 2 | 4 | 10 below |
| 3 | 9 | 5 below |
| 4 | 16 | 2 above |
| 5 | 25 | 11 above |
| 6 | 36 | 22 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 3), divide 14 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 14 ÷ x | New guess (x + 14÷x) ÷ 2 |
|---|---|---|---|
| 1 | 3 | 4.666667 | 3.833333 |
| 2 | 3.833333 | 3.652174 | 3.742754 |
| 3 | 3.742754 | 3.740561 | 3.741658 |
| 4 | 3.741658 | 3.741657 | 3.741657 |
After 4 passes the guess is 3.741657, already matching √14 = 3.741657 to six decimal places.
Exact form from the prime factors
14 = 2 × 7. No prime appears twice, so nothing can be pulled out of the root -- √14 is already in simplest form.
Frequently Asked Questions
What is the square root of 14?
The square root of 14 is approximately 3.741657, or exactly √14 in simplified radical form.
Is 14 a perfect square?
No, 14 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √14 between?
Between 3 and 4, because 3² = 9 and 4² = 16.
What are the square roots of the numbers right next to 14?
√13 is about 3.6056, and √15 is about 3.873. 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 √14 squared?
Exactly 14 -- squaring undoes the square root. Squaring the rounded decimal 3.741657 gives 14, which is why the radical form √14 is the exact answer and the decimal is an approximation.
What is the negative square root of 14?
−3.741657. Every positive number has two square roots, one positive and one negative, because (−3.741657)² is also 14. The radical sign √ by convention means the positive one.
More Number Theory
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