Square Root of 51
The square root of 51 is approximately 7.141428, or exactly √51 in simplified radical form. No, 51 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √51 sits
7² = 49 and 8² = 64, and 51 falls between them, so √51 is between 7 and 8. It's closer to 7 (2 above 49, 13 below 64), which matches the decimal 7.141428.
| k | k² | Compared to 51 |
|---|---|---|
| 5 | 25 | 26 below |
| 6 | 36 | 15 below |
| 7 | 49 | 2 below |
| 8 | 64 | 13 above |
| 9 | 81 | 30 above |
| 10 | 100 | 49 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 7), divide 51 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 51 ÷ x | New guess (x + 51÷x) ÷ 2 |
|---|---|---|---|
| 1 | 7 | 7.285714 | 7.142857 |
| 2 | 7.142857 | 7.14 | 7.141429 |
| 3 | 7.141429 | 7.141428 | 7.141428 |
| 4 | 7.141428 | 7.141428 | 7.141428 |
After 4 passes the guess is 7.141428, already matching √51 = 7.141428 to six decimal places.
Exact form from the prime factors
51 = 3 × 17. No prime appears twice, so nothing can be pulled out of the root -- √51 is already in simplest form.
Frequently Asked Questions
What is the square root of 51?
The square root of 51 is approximately 7.141428, or exactly √51 in simplified radical form.
Is 51 a perfect square?
No, 51 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √51 between?
Between 7 and 8, because 7² = 49 and 8² = 64.
What are the square roots of the numbers right next to 51?
√50 is about 7.0711, and √52 is about 7.2111. 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 √51 squared?
Exactly 51 -- squaring undoes the square root. Squaring the rounded decimal 7.141428 gives 51, which is why the radical form √51 is the exact answer and the decimal is an approximation.
What is the negative square root of 51?
−7.141428. Every positive number has two square roots, one positive and one negative, because (−7.141428)² is also 51. The radical sign √ by convention means the positive one.
More Number Theory
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