√921 at a glance
- Exact value
- √921
- Decimal (10 places)
- 30.3479818110
- Rounded
- 30.3 · 30.35 · 30.348
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.347982
- Prime factorization
- 3 × 307
- Cube root
- 9.729411
How to simplify √921
The prime factorization of 921 is 3 × 307. Every prime appears only once, so there is no pair to bring outside the radical — √921 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 921, 3 and 307 appear an odd number of times, so √921 is irrational and 30.3479818110 is a rounded value.
Where √921 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √921 lies between 30 and 31. 921 is 21 above 900 and 40 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.3443 (0.01% low)
- Tangent from 30, i.e. 30 + 21 ÷ 60: 30.3500 (0.01% high)
- Tangent from 31, i.e. 31 − 40 ÷ 62: 30.3548 (0.02% high)
For √921 the tangent at 30 wins, missing by only 0.002. Tangent estimates shine when the number sits close to a perfect square — here 921 is just 21 above 900.
Finding √921 with the Babylonian method
This is Newton’s method applied to f(x) = x² − 921: following the tangent line down to zero simplifies to averaging x with 921 ÷ x.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 921 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.7000000000 | 30.3500000000 | 2 |
| 2 | 30.3500000000 | 30.3459637562 | 30.3479818781 | 7 |
| 3 | 30.3479818781 | 30.3479817439 | 30.3479818110 | all 10 shown |
The count of correct decimals went 2, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √921 = 30.3479818110 to every decimal shown.
√921 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √921 the pattern is [30; 2, 1, 6, 1, 11, 3, 1, 2, 2, 3, 1, 1, …] with the block of 34 terms after the semicolon repeating forever (only the first 12 of the 34 are shown). A pattern that never ends is one more proof that √921 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 |
|---|---|---|
| 30/1 | 30.0000000000 | 3.5 × 10⁻¹ |
| 61/2 | 30.5000000000 | 1.5 × 10⁻¹ |
| 91/3 | 30.3333333333 | 1.5 × 10⁻² |
| 607/20 | 30.3500000000 | 2.0 × 10⁻³ |
| 698/23 | 30.3478260870 | 1.6 × 10⁻⁴ |
| 8,285/273 | 30.3479853480 | 3.5 × 10⁻⁶ |
The same fractions solve Pell’s equation, x² − 921y² = 1. Its smallest solution in positive whole numbers is x = 2,522,057,712,835,735, y = 83,104,627,139,412 — 16 digits for x, even though 921 is small, which is what makes Pell’s equation famous.
√921 in geometry and everyday measurements
- 921 square feet is 85.6 m². Laid out as a square — a small house footprint or a lot — it is about 30.35 ft (30 ft 4 in) on a side.
- 921 is not a sum of two whole-number squares — the prime factor 3 and 307 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √921 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 4 × 8 × 29 box, because 4² + 8² + 29² = 921.
Square roots near √921 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √918 | 3√102 | 30.2985 | No |
| √919 | √919 | 30.3150 | No |
| √920 | 2√230 | 30.3315 | No |
| √921 | √921 | 30.3480 | No |
| √922 | √922 | 30.3645 | No |
| √923 | √923 | 30.3809 | No |
| √924 | 2√231 | 30.3974 | No |
- The cube root of 921 is about 9.729411.
- Squaring undoes the root: (√921)² = 921, while 921² = 848,241 — the number whose square root is 921.
Frequently asked questions
What is the square root of 921?
The square root of 921 is √921, about 30.3479818110. The negative root, −30.347982, also squares to 921.
Is the square root of 921 rational or irrational?
Irrational. 921 is not a perfect square — it falls between 900 and 961 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √921 be simplified?
No. 921 = 3 × 307 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √921 rounded to two decimal places?
√921 ≈ 30.35 to two decimal places (30.3 to one, 30.348 to three). Check: 30.35² = 921.1225, close to 921.