Square Root of 18
The square root of 18 is approximately 4.242641, or exactly 3√2 in simplified radical form. No, 18 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √18 sits
4² = 16 and 5² = 25, and 18 falls between them, so √18 is between 4 and 5. It's closer to 4 (2 above 16, 7 below 25), which matches the decimal 4.242641.
| k | k² | Compared to 18 |
|---|---|---|
| 2 | 4 | 14 below |
| 3 | 9 | 9 below |
| 4 | 16 | 2 below |
| 5 | 25 | 7 above |
| 6 | 36 | 18 above |
| 7 | 49 | 31 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 18 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 18 ÷ x | New guess (x + 18÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 4.5 | 4.25 |
| 2 | 4.25 | 4.235294 | 4.242647 |
| 3 | 4.242647 | 4.242634 | 4.242641 |
| 4 | 4.242641 | 4.242641 | 4.242641 |
After 4 passes the guess is 4.242641, already matching √18 = 4.242641 to six decimal places.
Exact form from the prime factors
18 = 2 × 3^2. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √18 = 3√2. That's the exact value; the decimal 4.242641 is a rounding of it.
Frequently Asked Questions
What is the square root of 18?
The square root of 18 is approximately 4.242641, or exactly 3√2 in simplified radical form.
Is 18 a perfect square?
No, 18 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √18 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 18?
√17 is about 4.1231, and √19 is about 4.3589. 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 √18 squared?
Exactly 18 -- squaring undoes the square root. Squaring the rounded decimal 4.242641 gives 18, which is why the radical form 3√2 is the exact answer and the decimal is an approximation.
What is the negative square root of 18?
−4.242641. Every positive number has two square roots, one positive and one negative, because (−4.242641)² is also 18. The radical sign √ by convention means the positive one.
More Number Theory
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