√942 at a glance
- Exact value
- √942
- Decimal (10 places)
- 30.6920185064
- Rounded
- 30.7 · 30.69 · 30.692
- Perfect square?
- No — between 30² and 31²
- Rational?
- Irrational
- Both square roots
- ±30.692019
- Prime factorization
- 2 × 3 × 157
- Cube root
- 9.802804
How to simplify √942
The prime factorization of 942 is 2 × 3 × 157. Every prime appears only once, so there is no pair to bring outside the radical — √942 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 942, 2, 3 and 157 appear an odd number of times, so √942 is irrational and 30.6920185064 is a rounded value.
Where √942 sits between perfect squares
900 = 30² and 961 = 31² are the nearest perfect squares, so √942 lies between 30 and 31. 942 is 42 above 900 and 19 below 961, so the root is closer to 31.
- Straight line between 900 and 961: 30.6885 (0.01% low)
- Tangent from 30, i.e. 30 + 42 ÷ 60: 30.7000 (0.03% high)
- Tangent from 31, i.e. 31 − 19 ÷ 62: 30.6935 (0% high)
For √942 the tangent at 31 wins, missing by only 0.0015. Tangent estimates shine when the number sits close to a perfect square — here 942 is just 19 below 961.
Finding √942 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 | 942 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 31.0000000000 | 30.3870967742 | 30.6935483871 | 2 |
| 2 | 30.6935483871 | 30.6904887020 | 30.6920185446 | 7 |
| 3 | 30.6920185446 | 30.6920184683 | 30.6920185064 | 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 √942 = 30.6920185064 to every decimal shown.
√942 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √942 the pattern is [30; 1, 2, 4, 20, 4, 2, 1, 60] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √942 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 | 6.9 × 10⁻¹ |
| 31/1 | 31.0000000000 | 3.1 × 10⁻¹ |
| 92/3 | 30.6666666667 | 2.5 × 10⁻² |
| 399/13 | 30.6923076923 | 2.9 × 10⁻⁴ |
| 8,072/263 | 30.6920152091 | 3.3 × 10⁻⁶ |
| 32,687/1,065 | 30.6920187793 | 2.7 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 942y² = 1. Its smallest solution in positive whole numbers is x = 106,133, y = 3,458.
√942 in geometry and everyday measurements
- 942 square feet is 87.5 m². Laid out as a square — a small house footprint or a lot — it is about 30.69 ft (30 ft 8 in) on a side.
- 942 is not a sum of two whole-number squares — the prime factor 3 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √942 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 10 × 29 box, because 1² + 10² + 29² = 942.
Square roots near √942 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √939 | √939 | 30.6431 | No |
| √940 | 2√235 | 30.6594 | No |
| √941 | √941 | 30.6757 | No |
| √942 | √942 | 30.6920 | No |
| √943 | √943 | 30.7083 | No |
| √944 | 4√59 | 30.7246 | No |
| √945 | 3√105 | 30.7409 | No |
- The cube root of 942 is about 9.802804.
- Squaring undoes the root: (√942)² = 942, while 942² = 887,364 — the number whose square root is 942.
Frequently asked questions
What is the square root of 942?
The square root of 942 is √942, about 30.6920185064. The negative root, −30.692019, also squares to 942.
Is the square root of 942 rational or irrational?
Irrational. 942 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 √942 be simplified?
No. 942 = 2 × 3 × 157 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √942 rounded to two decimal places?
√942 ≈ 30.69 to two decimal places (30.7 to one, 30.692 to three). Check: 30.69² = 941.8761, close to 942.