Square Root of 19
The square root of 19 is approximately 4.358899, or exactly √19 in simplified radical form. No, 19 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √19 sits
4² = 16 and 5² = 25, and 19 falls between them, so √19 is between 4 and 5. It's closer to 4 (3 above 16, 6 below 25), which matches the decimal 4.358899.
| k | k² | Compared to 19 |
|---|---|---|
| 2 | 4 | 15 below |
| 3 | 9 | 10 below |
| 4 | 16 | 3 below |
| 5 | 25 | 6 above |
| 6 | 36 | 17 above |
| 7 | 49 | 30 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 19 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 19 ÷ x | New guess (x + 19÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 4.75 | 4.375 |
| 2 | 4.375 | 4.342857 | 4.358929 |
| 3 | 4.358929 | 4.358869 | 4.358899 |
| 4 | 4.358899 | 4.358899 | 4.358899 |
After 4 passes the guess is 4.358899, already matching √19 = 4.358899 to six decimal places.
Exact form from the prime factors
19 = 19. No prime appears twice, so nothing can be pulled out of the root -- √19 is already in simplest form.
Frequently Asked Questions
What is the square root of 19?
The square root of 19 is approximately 4.358899, or exactly √19 in simplified radical form.
Is 19 a perfect square?
No, 19 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √19 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 19?
√18 is about 4.2426, and √20 is about 4.4721. 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 √19 squared?
Exactly 19 -- squaring undoes the square root. Squaring the rounded decimal 4.358899 gives 19, which is why the radical form √19 is the exact answer and the decimal is an approximation.
What is the negative square root of 19?
−4.358899. Every positive number has two square roots, one positive and one negative, because (−4.358899)² is also 19. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
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- GCF Calculator
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