Square Root of 71
The square root of 71 is approximately 8.42615, or exactly √71 in simplified radical form. No, 71 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √71 sits
8² = 64 and 9² = 81, and 71 falls between them, so √71 is between 8 and 9. It's closer to 8 (7 above 64, 10 below 81), which matches the decimal 8.42615.
| k | k² | Compared to 71 |
|---|---|---|
| 6 | 36 | 35 below |
| 7 | 49 | 22 below |
| 8 | 64 | 7 below |
| 9 | 81 | 10 above |
| 10 | 100 | 29 above |
| 11 | 121 | 50 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 71 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 71 ÷ x | New guess (x + 71÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 8.875 | 8.4375 |
| 2 | 8.4375 | 8.414815 | 8.426157 |
| 3 | 8.426157 | 8.426142 | 8.42615 |
| 4 | 8.42615 | 8.42615 | 8.42615 |
After 4 passes the guess is 8.42615, already matching √71 = 8.42615 to six decimal places.
Exact form from the prime factors
71 = 71. No prime appears twice, so nothing can be pulled out of the root -- √71 is already in simplest form.
Frequently Asked Questions
What is the square root of 71?
The square root of 71 is approximately 8.42615, or exactly √71 in simplified radical form.
Is 71 a perfect square?
No, 71 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √71 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 71?
√70 is about 8.3666, and √72 is about 8.4853. 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 √71 squared?
Exactly 71 -- squaring undoes the square root. Squaring the rounded decimal 8.42615 gives 71, which is why the radical form √71 is the exact answer and the decimal is an approximation.
What is the negative square root of 71?
−8.42615. Every positive number has two square roots, one positive and one negative, because (−8.42615)² is also 71. The radical sign √ by convention means the positive one.
More Number Theory
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