√923 at a glance
- Exact value
- √923
- Decimal (10 places)
- 30.3809150619
- Rounded
- 30.4 · 30.38 · 30.381
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.380915
- Prime factorization
- 13 × 71
- Cube root
- 9.736448
How to simplify √923
The prime factorization of 923 is 13 × 71. Every prime appears only once, so there is no pair to bring outside the radical — √923 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 923, 13 and 71 appear an odd number of times, so √923 is irrational and 30.3809150619 is a rounded value.
Where √923 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √923 lies between 30 and 31. 923 is 23 above 900 and 38 below 961, so the root is closer to 30.
- Straight line between 900 and 961: 30.3770 (0.01% low)
- Tangent from 30, i.e. 30 + 23 ÷ 60: 30.3833 (0.01% high)
- Tangent from 31, i.e. 31 − 38 ÷ 62: 30.3871 (0.02% high)
For √923 the tangent at 30 wins, missing by only 0.0024. Tangent estimates shine when the number sits close to a perfect square — here 923 is just 23 above 900.
Finding √923 with the Babylonian method
If a guess is too big, 923 ÷ x is too small by about the same amount, so their average lands much closer. In fact an error of e shrinks to roughly e² ÷ (2√923) in one step.
Start from the nearest whole number, 30 (30² = 900):
| Step | Guess x | 923 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 30.0000000000 | 30.7666666667 | 30.3833333333 | 2 |
| 2 | 30.3833333333 | 30.3784969830 | 30.3809151582 | 7 |
| 3 | 30.3809151582 | 30.3809149657 | 30.3809150619 | 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 √923 = 30.3809150619 to every decimal shown.
√923 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √923 the pattern is [30; 2, 1, 1, 1, 2, 60] with the block of 6 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √923 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 | 3.8 × 10⁻¹ |
| 61/2 | 30.5000000000 | 1.2 × 10⁻¹ |
| 91/3 | 30.3333333333 | 4.8 × 10⁻² |
| 152/5 | 30.4000000000 | 1.9 × 10⁻² |
| 243/8 | 30.3750000000 | 5.9 × 10⁻³ |
| 638/21 | 30.3809523810 | 3.7 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 923y² = 1. Its smallest solution in positive whole numbers is x = 638, y = 21.
√923 in geometry and everyday measurements
- 923 square feet is 85.7 m². Laid out as a square — a small house footprint or a lot — it is about 30.38 ft (30 ft 5 in) on a side.
- 923 is not a sum of two whole-number squares — the prime factor 71 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √923 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 9 × 29 box, because 1² + 9² + 29² = 923.
Square roots near √923 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √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 |
| √925 | 5√37 | 30.4138 | No |
| √926 | √926 | 30.4302 | No |
- The cube root of 923 is about 9.736448.
- Squaring undoes the root: (√923)² = 923, while 923² = 851,929 — the number whose square root is 923.
Frequently asked questions
What is the square root of 923?
The square root of 923 is √923, about 30.3809150619. The negative root, −30.380915, also squares to 923.
Is the square root of 923 rational or irrational?
Irrational. 923 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 √923 be simplified?
No. 923 = 13 × 71 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √923 rounded to two decimal places?
√923 ≈ 30.38 to two decimal places (30.4 to one, 30.381 to three). Check: 30.38² = 922.9444, close to 923.