Square Root of 27
The square root of 27 is approximately 5.196152, or exactly 3√3 in simplified radical form. No, 27 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √27 sits
5² = 25 and 6² = 36, and 27 falls between them, so √27 is between 5 and 6. It's closer to 5 (2 above 25, 9 below 36), which matches the decimal 5.196152.
| k | k² | Compared to 27 |
|---|---|---|
| 3 | 9 | 18 below |
| 4 | 16 | 11 below |
| 5 | 25 | 2 below |
| 6 | 36 | 9 above |
| 7 | 49 | 22 above |
| 8 | 64 | 37 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 5), divide 27 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 27 ÷ x | New guess (x + 27÷x) ÷ 2 |
|---|---|---|---|
| 1 | 5 | 5.4 | 5.2 |
| 2 | 5.2 | 5.192308 | 5.196154 |
| 3 | 5.196154 | 5.196151 | 5.196152 |
| 4 | 5.196152 | 5.196152 | 5.196152 |
After 4 passes the guess is 5.196152, already matching √27 = 5.196152 to six decimal places.
Exact form from the prime factors
27 = 3^3. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √27 = 3√3. That's the exact value; the decimal 5.196152 is a rounding of it.
Frequently Asked Questions
What is the square root of 27?
The square root of 27 is approximately 5.196152, or exactly 3√3 in simplified radical form.
Is 27 a perfect square?
No, 27 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √27 between?
Between 5 and 6, because 5² = 25 and 6² = 36.
What are the square roots of the numbers right next to 27?
√26 is about 5.099, and √28 is about 5.2915. 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 √27 squared?
Exactly 27 -- squaring undoes the square root. Squaring the rounded decimal 5.196152 gives 27, which is why the radical form 3√3 is the exact answer and the decimal is an approximation.
What is the negative square root of 27?
−5.196152. Every positive number has two square roots, one positive and one negative, because (−5.196152)² is also 27. The radical sign √ by convention means the positive one.
More Number Theory
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