√946 at a glance
- Exact value
- √946
- Decimal (10 places)
- 30.7571129985
- Rounded
- 30.8 · 30.76 · 30.757
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.757113
- Prime factorization
- 2 × 11 × 43
- Cube root
- 9.816659
How to simplify √946
The prime factorization of 946 is 2 × 11 × 43. Every prime appears only once, so there is no pair to bring outside the radical — √946 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 946, 2, 11 and 43 appear an odd number of times, so √946 is irrational and 30.7571129985 is a rounded value.
Where √946 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √946 lies between 30 and 31. 946 is 46 above 900 and 15 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.7541 (0.01% low)
- Tangent from 30, i.e. 30 + 46 ÷ 60: 30.7667 (0.03% high)
- Tangent from 31, i.e. 31 − 15 ÷ 62: 30.7581 (0% high)
For √946 the tangent at 31 wins, missing by only 0.001. Tangent estimates shine when the number sits close to a perfect square — here 946 is just 15 below 961.
Finding √946 with the Babylonian method
The idea is ancient. A Babylonian clay tablet catalogued as YBC 7289, about 3,800 years old, records √2 correct to roughly six decimal places, and Heron of Alexandria described this averaging rule in the first century AD.
Start from the nearest whole number, 31 (31² = 961):
| Step | Guess x | 946 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.5161290323 | 30.7580645161 | 3 |
| 2 | 30.7580645161 | 30.7561615102 | 30.7571130132 | 7 |
| 3 | 30.7571130132 | 30.7571129837 | 30.7571129985 | all 10 shown |
The count of correct decimals went 3, 7 and all 10 over 3 steps — roughly doubling each time — until the guess matched √946 = 30.7571129985 to every decimal shown.
√946 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √946 the pattern is [30; 1, 3, 8, 1, 1, 6, 3, 3, 1, 3, 1, 1, …] with the block of 30 terms after the semicolon repeating forever (only the first 12 of the 30 are shown). A pattern that never ends is one more proof that √946 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 | 7.6 × 10⁻¹ |
| 31/1 | 31.0000000000 | 2.4 × 10⁻¹ |
| 123/4 | 30.7500000000 | 7.1 × 10⁻³ |
| 1,015/33 | 30.7575757576 | 4.6 × 10⁻⁴ |
| 1,138/37 | 30.7567567568 | 3.6 × 10⁻⁴ |
| 2,153/70 | 30.7571428571 | 3.0 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 946y² = 1. Its smallest solution in positive whole numbers is x = 45,225,786,400,145, y = 1,470,417,148,788 — 14 digits for x, even though 946 is small, which is what makes Pell’s equation famous.
√946 in geometry and everyday measurements
- 946 square feet is 87.9 m². Laid out as a square — a small house footprint or a lot — it is about 30.76 ft (30 ft 9 in) on a side.
- 946 is not a sum of two whole-number squares — the prime factor 11 and 43 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √946 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 3 × 19 × 24 box, because 3² + 19² + 24² = 946.
Square roots near √946 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √943 | √943 | 30.7083 | No |
| √944 | 4√59 | 30.7246 | No |
| √945 | 3√105 | 30.7409 | No |
| √946 | √946 | 30.7571 | No |
| √947 | √947 | 30.7734 | No |
| √948 | 2√237 | 30.7896 | No |
| √949 | √949 | 30.8058 | No |
- The cube root of 946 is about 9.816659.
- Squaring undoes the root: (√946)² = 946, while 946² = 894,916 — the number whose square root is 946.
Frequently asked questions
What is the square root of 946?
The square root of 946 is √946, about 30.7571129985. The negative root, −30.757113, also squares to 946.
Is the square root of 946 rational or irrational?
Irrational. 946 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 √946 be simplified?
No. 946 = 2 × 11 × 43 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √946 rounded to two decimal places?
√946 ≈ 30.76 to two decimal places (30.8 to one, 30.757 to three). Check: 30.76² = 946.1776, close to 946.