√46 at a glance
- Exact value
- √46
- Decimal (10 places)
- 6.7823299831
- Rounded
- 6.8 · 6.78 · 6.782
- Perfect square?
- No — between 6² and 7²
- Rational?
- Irrational
- Both square roots
- ±6.782330
- Prime factorization
- 2 × 23
- Cube root
- 3.583048
How to simplify √46
The prime factorization of 46 is 2 × 23. Every prime appears only once, so there is no pair to bring outside the radical — √46 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 46, 2 and 23 appear an odd number of times, so √46 is irrational and 6.7823299831 is a rounded value.
Where √46 sits between perfect squares
36 = 6² and 49 = 7² are the nearest perfect squares, so √46 lies between 6 and 7. 46 is 10 above 36 and 3 below 49, so the root is closer to 7.
- Straight line between 36 and 49: 6.7692 (0.19% low)
- Tangent from 6, i.e. 6 + 10 ÷ 12: 6.8333 (0.75% high)
- Tangent from 7, i.e. 7 − 3 ÷ 14: 6.7857 (0.05% high)
For √46 the tangent at 7 wins, missing by only 0.0034. Tangent estimates shine when the number sits close to a perfect square — here 46 is just 3 below 49.
Finding √46 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, 7 (7² = 49):
| Step | Guess x | 46 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 7.0000000000 | 6.5714285714 | 6.7857142857 | 2 |
| 2 | 6.7857142857 | 6.7789473684 | 6.7823308271 | 6 |
| 3 | 6.7823308271 | 6.7823291392 | 6.7823299831 | all 10 shown |
The count of correct decimals went 2, 6 and all 10 over 3 steps — roughly doubling each time — until the guess matched √46 = 6.7823299831 to every decimal shown.
√46 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √46 the pattern is [6; 1, 3, 1, 1, 2, 6, 2, 1, 1, 3, 1, 12] with the block of 12 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √46 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 |
|---|---|---|
| 6/1 | 6.0000000000 | 7.8 × 10⁻¹ |
| 7/1 | 7.0000000000 | 2.2 × 10⁻¹ |
| 27/4 | 6.7500000000 | 3.2 × 10⁻² |
| 34/5 | 6.8000000000 | 1.8 × 10⁻² |
| 61/9 | 6.7777777778 | 4.6 × 10⁻³ |
| 156/23 | 6.7826086957 | 2.8 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 46y² = 1. Its smallest solution in positive whole numbers is x = 24,335, y = 3,588.
√46 in geometry and everyday measurements
- A square room or garden bed covering 46 square feet measures about 6.78 ft (6 ft 9 in) along each wall.
- 46 is not a sum of two whole-number squares — the prime factor 23 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √46 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 3 × 6 box, because 1² + 3² + 6² = 46.
Square roots near √46 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √43 | √43 | 6.5574 | No |
| √44 | 2√11 | 6.6332 | No |
| √45 | 3√5 | 6.7082 | No |
| √46 | √46 | 6.7823 | No |
| √47 | √47 | 6.8557 | No |
| √48 | 4√3 | 6.9282 | No |
| √49 | 7 | 7.0000 | Yes |
- The cube root of 46 is about 3.583048.
- Four times the radicand doubles the root: √184 = 2 × √46 ≈ 13.56466.
Frequently asked questions
What is the square root of 46?
The square root of 46 is √46, about 6.7823299831. The negative root, −6.782330, also squares to 46.
Is the square root of 46 rational or irrational?
Irrational. 46 is not a perfect square — it falls between 36 and 49 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √46 be simplified?
No. 46 = 2 × 23 has no repeated prime factor, so there is no perfect square to take out of the radical.
What is √46 rounded to two decimal places?
√46 ≈ 6.78 to two decimal places (6.8 to one, 6.782 to three). Check: 6.78² = 45.9684, close to 46.