Square Root of 73
The square root of 73 is approximately 8.544004, or exactly √73 in simplified radical form. No, 73 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √73 sits
8² = 64 and 9² = 81, and 73 falls between them, so √73 is between 8 and 9. It's closer to 9 (9 above 64, 8 below 81), which matches the decimal 8.544004.
| k | k² | Compared to 73 |
|---|---|---|
| 6 | 36 | 37 below |
| 7 | 49 | 24 below |
| 8 | 64 | 9 below |
| 9 | 81 | 8 above |
| 10 | 100 | 27 above |
| 11 | 121 | 48 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 73 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 73 ÷ x | New guess (x + 73÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9.125 | 8.5625 |
| 2 | 8.5625 | 8.525547 | 8.544024 |
| 3 | 8.544024 | 8.543984 | 8.544004 |
| 4 | 8.544004 | 8.544004 | 8.544004 |
After 4 passes the guess is 8.544004, already matching √73 = 8.544004 to six decimal places.
Exact form from the prime factors
73 = 73. No prime appears twice, so nothing can be pulled out of the root -- √73 is already in simplest form.
Frequently Asked Questions
What is the square root of 73?
The square root of 73 is approximately 8.544004, or exactly √73 in simplified radical form.
Is 73 a perfect square?
No, 73 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √73 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 73?
√72 is about 8.4853, and √74 is about 8.6023. 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 √73 squared?
Exactly 73 -- squaring undoes the square root. Squaring the rounded decimal 8.544004 gives 73, which is why the radical form √73 is the exact answer and the decimal is an approximation.
What is the negative square root of 73?
−8.544004. Every positive number has two square roots, one positive and one negative, because (−8.544004)² is also 73. The radical sign √ by convention means the positive one.
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