Square Root of 80
The square root of 80 is approximately 8.944272, or exactly 4√5 in simplified radical form. No, 80 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √80 sits
8² = 64 and 9² = 81, and 80 falls between them, so √80 is between 8 and 9. It's closer to 9 (16 above 64, 1 below 81), which matches the decimal 8.944272.
| k | k² | Compared to 80 |
|---|---|---|
| 6 | 36 | 44 below |
| 7 | 49 | 31 below |
| 8 | 64 | 16 below |
| 9 | 81 | 1 above |
| 10 | 100 | 20 above |
| 11 | 121 | 41 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 80 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 80 ÷ x | New guess (x + 80÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 10 | 9 |
| 2 | 9 | 8.888889 | 8.944444 |
| 3 | 8.944444 | 8.944099 | 8.944272 |
| 4 | 8.944272 | 8.944272 | 8.944272 |
After 4 passes the guess is 8.944272, already matching √80 = 8.944272 to six decimal places.
Exact form from the prime factors
80 = 2^4 × 5. Each pair of matching primes comes out of the root as one factor, and what's left stays inside: √80 = 4√5. That's the exact value; the decimal 8.944272 is a rounding of it.
Frequently Asked Questions
What is the square root of 80?
The square root of 80 is approximately 8.944272, or exactly 4√5 in simplified radical form.
Is 80 a perfect square?
No, 80 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √80 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 80?
√79 is about 8.8882, and √81 is exactly 9. 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 √80 squared?
Exactly 80 -- squaring undoes the square root. Squaring the rounded decimal 8.944272 gives 80, which is why the radical form 4√5 is the exact answer and the decimal is an approximation.
What is the negative square root of 80?
−8.944272. Every positive number has two square roots, one positive and one negative, because (−8.944272)² is also 80. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 80
- GCF Calculator
- LCM Calculator