Square Root of 70
The square root of 70 is approximately 8.3666, or exactly √70 in simplified radical form. No, 70 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √70 sits
8² = 64 and 9² = 81, and 70 falls between them, so √70 is between 8 and 9. It's closer to 8 (6 above 64, 11 below 81), which matches the decimal 8.3666.
| k | k² | Compared to 70 |
|---|---|---|
| 6 | 36 | 34 below |
| 7 | 49 | 21 below |
| 8 | 64 | 6 below |
| 9 | 81 | 11 above |
| 10 | 100 | 30 above |
| 11 | 121 | 51 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 8), divide 70 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 70 ÷ x | New guess (x + 70÷x) ÷ 2 |
|---|---|---|---|
| 1 | 8 | 8.75 | 8.375 |
| 2 | 8.375 | 8.358209 | 8.366604 |
| 3 | 8.366604 | 8.366596 | 8.3666 |
| 4 | 8.3666 | 8.3666 | 8.3666 |
After 4 passes the guess is 8.3666, already matching √70 = 8.3666 to six decimal places.
Exact form from the prime factors
70 = 2 × 5 × 7. No prime appears twice, so nothing can be pulled out of the root -- √70 is already in simplest form.
Frequently Asked Questions
What is the square root of 70?
The square root of 70 is approximately 8.3666, or exactly √70 in simplified radical form.
Is 70 a perfect square?
No, 70 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √70 between?
Between 8 and 9, because 8² = 64 and 9² = 81.
What are the square roots of the numbers right next to 70?
√69 is about 8.3066, and √71 is about 8.4261. 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 √70 squared?
Exactly 70 -- squaring undoes the square root. Squaring the rounded decimal 8.3666 gives 70, which is why the radical form √70 is the exact answer and the decimal is an approximation.
What is the negative square root of 70?
−8.3666. Every positive number has two square roots, one positive and one negative, because (−8.3666)² is also 70. The radical sign √ by convention means the positive one.
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