√94 at a glance
- Exact value
- √94
- Decimal (10 places)
- 9.6953597148
- Rounded
- 9.7 · 9.70 · 9.695
- Perfect square?
- No — between 9² and 10²
- Rational?
- Irrational
- Both square roots
- ±9.695360
- Prime factorization
- 2 × 47
- Cube root
- 4.546836
How to simplify √94
The prime factorization of 94 is 2 × 47. Every prime appears only once, so there is no pair to bring outside the radical — √94 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 94, 2 and 47 appear an odd number of times, so √94 is irrational and 9.6953597148 is a rounded value.
Where √94 sits between perfect squares
81 = 9² and 100 = 10² are the nearest perfect squares, so √94 lies between 9 and 10. 94 is 13 above 81 and 6 below 100, so the root is closer to 10.
- Straight line between 81 and 100: 9.6842 (0.11% low)
- Tangent from 9, i.e. 9 + 13 ÷ 18: 9.7222 (0.28% high)
- Tangent from 10, i.e. 10 − 6 ÷ 20: 9.7000 (0.05% high)
For √94 the tangent at 10 wins, missing by only 0.0046. Tangent estimates shine when the number sits close to a perfect square — here 94 is just 6 below 100.
Finding √94 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, 10 (10² = 100):
| Step | Guess x | 94 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 10.0000000000 | 9.4000000000 | 9.7000000000 | 2 |
| 2 | 9.7000000000 | 9.6907216495 | 9.6953608247 | 5 |
| 3 | 9.6953608247 | 9.6953586049 | 9.6953597148 | all 10 shown |
The count of correct decimals went 2, 5 and all 10 over 3 steps — roughly doubling each time — until the guess matched √94 = 9.6953597148 to every decimal shown.
√94 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √94 the pattern is [9; 1, 2, 3, 1, 1, 5, 1, 8, 1, 5, 1, 1, …] with the block of 16 terms after the semicolon repeating forever (only the first 12 of the 16 are shown). A pattern that never ends is one more proof that √94 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 |
|---|---|---|
| 9/1 | 9.0000000000 | 7.0 × 10⁻¹ |
| 10/1 | 10.0000000000 | 3.0 × 10⁻¹ |
| 29/3 | 9.6666666667 | 2.9 × 10⁻² |
| 97/10 | 9.7000000000 | 4.6 × 10⁻³ |
| 126/13 | 9.6923076923 | 3.1 × 10⁻³ |
| 223/23 | 9.6956521739 | 2.9 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 94y² = 1. Its smallest solution in positive whole numbers is x = 2,143,295, y = 221,064.
√94 in geometry and everyday measurements
- A square room or garden bed covering 94 square feet measures about 9.7 ft (9 ft 8 in) along each wall.
- 94 is not a sum of two whole-number squares — the prime factor 47 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √94 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 2 × 3 × 9 box, because 2² + 3² + 9² = 94.
Square roots near √94 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √91 | √91 | 9.5394 | No |
| √92 | 2√23 | 9.5917 | No |
| √93 | √93 | 9.6437 | No |
| √94 | √94 | 9.6954 | No |
| √95 | √95 | 9.7468 | No |
| √96 | 4√6 | 9.7980 | No |
| √97 | √97 | 9.8489 | No |
- The cube root of 94 is about 4.546836.
- Four times the radicand doubles the root: √376 = 2 × √94 ≈ 19.390719.
Frequently asked questions
What is the square root of 94?
The square root of 94 is √94, about 9.6953597148. The negative root, −9.695360, also squares to 94.
Is the square root of 94 rational or irrational?
Irrational. 94 is not a perfect square — it falls between 81 and 100 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √94 be simplified?
No. 94 = 2 × 47 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √94 rounded to two decimal places?
√94 ≈ 9.70 to two decimal places (9.7 to one, 9.695 to three). Check: 9.70² = 94.09, close to 94.