Square Root of 74
The square root of 74 is approximately 8.602325, or exactly √74 in simplified radical form. No, 74 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √74 sits
8² = 64 and 9² = 81, and 74 falls between them, so √74 is between 8 and 9. It's closer to 9 (10 above 64, 7 below 81), which matches the decimal 8.602325.
| k | k² | Compared to 74 |
|---|---|---|
| 6 | 36 | 38 below |
| 7 | 49 | 25 below |
| 8 | 64 | 10 below |
| 9 | 81 | 7 above |
| 10 | 100 | 26 above |
| 11 | 121 | 47 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 74 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 74 ÷ x | New guess (x + 74÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 9.25 | 8.625 |
| 2 | 8.625 | 8.57971 | 8.602355 |
| 3 | 8.602355 | 8.602295 | 8.602325 |
| 4 | 8.602325 | 8.602325 | 8.602325 |
After 4 passes the guess is 8.602325, already matching √74 = 8.602325 to six decimal places.
Exact form from the prime factors
74 = 2 × 37. No prime appears twice, so nothing can be pulled out of the root -- √74 is already in simplest form.
Frequently Asked Questions
What is the square root of 74?
The square root of 74 is approximately 8.602325, or exactly √74 in simplified radical form.
Is 74 a perfect square?
No, 74 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √74 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 74?
√73 is about 8.544, and √75 is about 8.6603. 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 √74 squared?
Exactly 74 -- squaring undoes the square root. Squaring the rounded decimal 8.602325 gives 74, which is why the radical form √74 is the exact answer and the decimal is an approximation.
What is the negative square root of 74?
−8.602325. Every positive number has two square roots, one positive and one negative, because (−8.602325)² is also 74. The radical sign √ by convention means the positive one.
More Number Theory
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