√21 at a glance
- Exact value
- √21
- Decimal (10 places)
- 4.5825756950
- Rounded
- 4.6 · 4.58 · 4.583
- Perfect square?
- No — between 4² and 5²
- Rational?
- Irrational
- Both square roots
- ±4.582576
- Prime factorization
- 3 × 7
- Cube root
- 2.758924
How to simplify √21
The prime factorization of 21 is 3 × 7. Every prime appears only once, so there is no pair to bring outside the radical — √21 is already in simplest form (a "square-free" radicand).
A whole number has a rational square root only when every prime in its factorization appears an even number of times. In 21, 3 and 7 appear an odd number of times, so √21 is irrational and 4.5825756950 is a rounded value.
Where √21 sits between perfect squares
16 = 4² and 25 = 5² are the nearest perfect squares, so √21 lies between 4 and 5. 21 is 5 above 16 and 4 below 25, so the root is closer to 5.
- Straight line between 16 and 25: 4.5556 (0.59% low)
- Tangent from 4, i.e. 4 + 5 ÷ 8: 4.6250 (0.93% high)
- Tangent from 5, i.e. 5 − 4 ÷ 10: 4.6000 (0.38% high)
For √21 the tangent at 5 wins, missing by only 0.0174. Tangent estimates shine when the number sits close to a perfect square — here 21 is just 4 below 25.
Finding √21 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 21: following the tangent line down to zero simplifies to averaging x with 21 ÷ x.
Start from the nearest whole number, 5 (5² = 25):
| Step | Guess x | 21 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 4.2000000000 | 4.6000000000 | 1 |
| 2 | 4.6000000000 | 4.5652173913 | 4.5826086957 | 4 |
| 3 | 4.5826086957 | 4.5825426945 | 4.5825756951 | 9 |
| 4 | 4.5825756951 | 4.5825756948 | 4.5825756950 | all 10 shown |
The count of correct decimals went 1, 4, 9 and all 10 over 4 steps — roughly doubling each time — until the guess matched √21 = 4.5825756950 to every decimal shown.
√21 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √21 the pattern is [4; 1, 1, 2, 1, 1, 8] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √21 is irrational.
Cutting the continued fraction off early gives the best fractions for approximating the root — no fraction with a smaller denominator comes closer:
| Fraction | Decimal | Error |
|---|---|---|
| 4/1 | 4.0000000000 | 5.8 × 10⁻¹ |
| 5/1 | 5.0000000000 | 4.2 × 10⁻¹ |
| 9/2 | 4.5000000000 | 8.3 × 10⁻² |
| 23/5 | 4.6000000000 | 1.7 × 10⁻² |
| 32/7 | 4.5714285714 | 1.1 × 10⁻² |
| 55/12 | 4.5833333333 | 7.6 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 21y² = 1. Its smallest solution in positive whole numbers is x = 55, y = 12.
√21 in geometry and everyday measurements
- A square tile with an area of 21 square inches has sides about 4.583 in long.
- 21 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √21 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 2 × 4 box, because 1² + 2² + 4² = 21.
Square roots near √21 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √18 | 3√2 | 4.2426 | No |
| √19 | √19 | 4.3589 | No |
| √20 | 2√5 | 4.4721 | No |
| √21 | √21 | 4.5826 | No |
| √22 | √22 | 4.6904 | No |
| √23 | √23 | 4.7958 | No |
| √24 | 2√6 | 4.8990 | No |
- The cube root of 21 is about 2.758924.
- Four times the radicand doubles the root: √84 = 2 × √21 ≈ 9.165151.
Frequently asked questions
What is the square root of 21?
The square root of 21 is √21, about 4.5825756950. The negative root, −4.582576, also squares to 21.
Is the square root of 21 rational or irrational?
Irrational. 21 is not a perfect square — it falls between 16 and 25 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √21 be simplified?
No. 21 = 3 × 7 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √21 rounded to two decimal places?
√21 ≈ 4.58 to two decimal places (4.6 to one, 4.583 to three). Check: 4.58² = 20.9764, close to 21.