Square Root of 72
The square root of 72 is approximately 8.485281, or exactly 6√2 in simplified radical form. No, 72 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √72 sits
8² = 64 and 9² = 81, and 72 falls between them, so √72 is between 8 and 9. It's closer to 8 (8 above 64, 9 below 81), which matches the decimal 8.485281.
| k | k² | Compared to 72 |
|---|---|---|
| 6 | 36 | 36 below |
| 7 | 49 | 23 below |
| 8 | 64 | 8 below |
| 9 | 81 | 9 above |
| 10 | 100 | 28 above |
| 11 | 121 | 49 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 72 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 72 ÷ x | New guess (x + 72÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9 | 8.5 |
| 2 | 8.5 | 8.470588 | 8.485294 |
| 3 | 8.485294 | 8.485269 | 8.485281 |
| 4 | 8.485281 | 8.485281 | 8.485281 |
After 4 passes the guess is 8.485281, already matching √72 = 8.485281 to six decimal places.
Exact form from the prime factors
72 = 2^3 × 3^2. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √72 = 6√2. That's the exact value; the decimal 8.485281 is a rounding of it.
Frequently Asked Questions
What is the square root of 72?
The square root of 72 is approximately 8.485281, or exactly 6√2 in simplified radical form.
Is 72 a perfect square?
No, 72 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √72 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 72?
√71 is about 8.4261, and √73 is about 8.544. 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 √72 squared?
Exactly 72 -- squaring undoes the square root. Squaring the rounded decimal 8.485281 gives 72, which is why the radical form 6√2 is the exact answer and the decimal is an approximation.
What is the negative square root of 72?
−8.485281. Every positive number has two square roots, one positive and one negative, because (−8.485281)² is also 72. The radical sign √ by convention means the positive one.
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