Square Root of 10
The square root of 10 is approximately 3.162278, or exactly √10 in simplified radical form. No, 10 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √10 sits
3² = 9 and 4² = 16, and 10 falls between them, so √10 is between 3 and 4. It's closer to 3 (1 above 9, 6 below 16), which matches the decimal 3.162278.
| k | k² | Compared to 10 |
|---|---|---|
| 1 | 1 | 9 below |
| 2 | 4 | 6 below |
| 3 | 9 | 1 below |
| 4 | 16 | 6 above |
| 5 | 25 | 15 above |
| 6 | 36 | 26 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 3), divide 10 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 10 ÷ x | New guess (x + 10÷x) ÷ 2 |
|---|---|---|---|
| 1 | 3 | 3.333333 | 3.166667 |
| 2 | 3.166667 | 3.157895 | 3.162281 |
| 3 | 3.162281 | 3.162275 | 3.162278 |
| 4 | 3.162278 | 3.162278 | 3.162278 |
After 4 passes the guess is 3.162278, already matching √10 = 3.162278 to six decimal places.
Exact form from the prime factors
10 = 2 × 5. No prime appears twice, so nothing can be pulled out of the root -- √10 is already in simplest form.
Frequently Asked Questions
What is the square root of 10?
The square root of 10 is approximately 3.162278, or exactly √10 in simplified radical form.
Is 10 a perfect square?
No, 10 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √10 between?
Between 3 and 4, because 3² = 9 and 4² = 16.
What are the square roots of the numbers right next to 10?
√9 is exactly 3, and √11 is about 3.3166. 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 √10 squared?
Exactly 10 -- squaring undoes the square root. Squaring the rounded decimal 3.162278 gives 10, which is why the radical form √10 is the exact answer and the decimal is an approximation.
What is the negative square root of 10?
−3.162278. Every positive number has two square roots, one positive and one negative, because (−3.162278)² is also 10. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 10
- GCF Calculator
- LCM Calculator