Square Root of 54
The square root of 54 is approximately 7.348469, or exactly 3√6 in simplified radical form. No, 54 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √54 sits
7² = 49 and 8² = 64, and 54 falls between them, so √54 is between 7 and 8. It's closer to 7 (5 above 49, 10 below 64), which matches the decimal 7.348469.
| k | k² | Compared to 54 |
|---|---|---|
| 5 | 25 | 29 below |
| 6 | 36 | 18 below |
| 7 | 49 | 5 below |
| 8 | 64 | 10 above |
| 9 | 81 | 27 above |
| 10 | 100 | 46 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 54 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 54 ÷ x | New guess (x + 54÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 7.714286 | 7.357143 |
| 2 | 7.357143 | 7.339806 | 7.348474 |
| 3 | 7.348474 | 7.348464 | 7.348469 |
| 4 | 7.348469 | 7.348469 | 7.348469 |
After 4 passes the guess is 7.348469, already matching √54 = 7.348469 to six decimal places.
Exact form from the prime factors
54 = 2 × 3^3. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √54 = 3√6. That's the exact value; the decimal 7.348469 is a rounding of it.
Frequently Asked Questions
What is the square root of 54?
The square root of 54 is approximately 7.348469, or exactly 3√6 in simplified radical form.
Is 54 a perfect square?
No, 54 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √54 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 54?
√53 is about 7.2801, and √55 is about 7.4162. 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 √54 squared?
Exactly 54 -- squaring undoes the square root. Squaring the rounded decimal 7.348469 gives 54, which is why the radical form 3√6 is the exact answer and the decimal is an approximation.
What is the negative square root of 54?
−7.348469. Every positive number has two square roots, one positive and one negative, because (−7.348469)² is also 54. The radical sign √ by convention means the positive one.
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