Square Root of 11
The square root of 11 is approximately 3.316625, or exactly √11 in simplified radical form. No, 11 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √11 sits
3² = 9 and 4² = 16, and 11 falls between them, so √11 is between 3 and 4. It's closer to 3 (2 above 9, 5 below 16), which matches the decimal 3.316625.
| k | k² | Compared to 11 |
|---|---|---|
| 1 | 1 | 10 below |
| 2 | 4 | 7 below |
| 3 | 9 | 2 below |
| 4 | 16 | 5 above |
| 5 | 25 | 14 above |
| 6 | 36 | 25 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 3), divide 11 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 11 ÷ x | New guess (x + 11÷x) ÷ 2 |
|---|---|---|---|
| 1 | 3 | 3.666667 | 3.333333 |
| 2 | 3.333333 | 3.3 | 3.316667 |
| 3 | 3.316667 | 3.316583 | 3.316625 |
| 4 | 3.316625 | 3.316625 | 3.316625 |
After 4 passes the guess is 3.316625, already matching √11 = 3.316625 to six decimal places.
Exact form from the prime factors
11 = 11. No prime appears twice, so nothing can be pulled out of the root -- √11 is already in simplest form.
Frequently Asked Questions
What is the square root of 11?
The square root of 11 is approximately 3.316625, or exactly √11 in simplified radical form.
Is 11 a perfect square?
No, 11 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √11 between?
Between 3 and 4, because 3² = 9 and 4² = 16.
What are the square roots of the numbers right next to 11?
√10 is about 3.1623, and √12 is about 3.4641. 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 √11 squared?
Exactly 11 -- squaring undoes the square root. Squaring the rounded decimal 3.316625 gives 11, which is why the radical form √11 is the exact answer and the decimal is an approximation.
What is the negative square root of 11?
−3.316625. Every positive number has two square roots, one positive and one negative, because (−3.316625)² is also 11. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 11
- GCF Calculator
- LCM Calculator