Square Root of 22
The square root of 22 is approximately 4.690416, or exactly √22 in simplified radical form. No, 22 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √22 sits
4² = 16 and 5² = 25, and 22 falls between them, so √22 is between 4 and 5. It's closer to 5 (6 above 16, 3 below 25), which matches the decimal 4.690416.
| k | k² | Compared to 22 |
|---|---|---|
| 2 | 4 | 18 below |
| 3 | 9 | 13 below |
| 4 | 16 | 6 below |
| 5 | 25 | 3 above |
| 6 | 36 | 14 above |
| 7 | 49 | 27 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 22 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 22 ÷ x | New guess (x + 22÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 5.5 | 4.75 |
| 2 | 4.75 | 4.631579 | 4.690789 |
| 3 | 4.690789 | 4.690042 | 4.690416 |
| 4 | 4.690416 | 4.690416 | 4.690416 |
After 4 passes the guess is 4.690416, already matching √22 = 4.690416 to six decimal places.
Exact form from the prime factors
22 = 2 × 11. No prime appears twice, so nothing can be pulled out of the root -- √22 is already in simplest form.
Frequently Asked Questions
What is the square root of 22?
The square root of 22 is approximately 4.690416, or exactly √22 in simplified radical form.
Is 22 a perfect square?
No, 22 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √22 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 22?
√21 is about 4.5826, and √23 is about 4.7958. 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 √22 squared?
Exactly 22 -- squaring undoes the square root. Squaring the rounded decimal 4.690416 gives 22, which is why the radical form √22 is the exact answer and the decimal is an approximation.
What is the negative square root of 22?
−4.690416. Every positive number has two square roots, one positive and one negative, because (−4.690416)² is also 22. The radical sign √ by convention means the positive one.
More Number Theory
- Square Root Calculator (look up any number)
- Factors of 22
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