Square Root of 21
The square root of 21 is approximately 4.582576, or exactly √21 in simplified radical form. No, 21 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
Where √21 sits
4² = 16 and 5² = 25, and 21 falls between them, so √21 is between 4 and 5. It's closer to 5 (5 above 16, 4 below 25), which matches the decimal 4.582576.
| k | k² | Compared to 21 |
|---|---|---|
| 2 | 4 | 17 below |
| 3 | 9 | 12 below |
| 4 | 16 | 5 below |
| 5 | 25 | 4 above |
| 6 | 36 | 15 above |
| 7 | 49 | 28 above |
Working it out by hand: Newton's method
Start with a guess (the nearest whole root, 4), divide 21 by it, and average the guess with the result. Each pass roughly doubles the number of correct digits:
| Pass | Guess x | 21 ÷ x | New guess (x + 21÷x) ÷ 2 |
|---|---|---|---|
| 1 | 4 | 5.25 | 4.625 |
| 2 | 4.625 | 4.540541 | 4.58277 |
| 3 | 4.58277 | 4.582381 | 4.582576 |
| 4 | 4.582576 | 4.582576 | 4.582576 |
After 4 passes the guess is 4.582576, already matching √21 = 4.582576 to six decimal places.
Exact form from the prime factors
21 = 3 × 7. No prime appears twice, so nothing can be pulled out of the root -- √21 is already in simplest form.
Frequently Asked Questions
What is the square root of 21?
The square root of 21 is approximately 4.582576, or exactly √21 in simplified radical form.
Is 21 a perfect square?
No, 21 is not a perfect square. Its square root is irrational and never terminates or repeats as a decimal.
What two whole numbers is √21 between?
Between 4 and 5, because 4² = 16 and 5² = 25.
What are the square roots of the numbers right next to 21?
√20 is about 4.4721, and √22 is about 4.6904. 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 √21 squared?
Exactly 21 -- squaring undoes the square root. Squaring the rounded decimal 4.582576 gives 21, which is why the radical form √21 is the exact answer and the decimal is an approximation.
What is the negative square root of 21?
−4.582576. Every positive number has two square roots, one positive and one negative, because (−4.582576)² is also 21. The radical sign √ by convention means the positive one.
More Number Theory
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