√378 at a glance
- Exact value
- 3√42
- Decimal (10 places)
- 19.4422220952
- Rounded
- 19.4 · 19.44 · 19.442
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.442222
- Prime factorization
- 2 × 3³ × 7
- Cube root
- 7.230427
How to simplify √378
Look for the largest perfect square that divides 378. Here it is 9 (3²), because 378 = 9 × 42 and 42 has no square factor left:
The prime factorization tells the same story: 378 = 2 × 3³ × 7. Each pair of equal primes leaves the radical as one factor, so 3 comes out and 2 × 3 × 7 stays inside.
Check: (3√42)² = 3² × 42 = 9 × 42 = 378. As a decimal, 3√42 = 3 × 6.4807406984 ≈ 19.4422220952.
Where √378 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √378 lies between 19 and 20. 378 is 17 above 361 and 22 below 400, so the root is closer to 19.
- Straight line between 361 and 400: 19.4359 (0.03% low)
- Tangent from 19, i.e. 19 + 17 ÷ 38: 19.4474 (0.03% high)
- Tangent from 20, i.e. 20 − 22 ÷ 40: 19.4500 (0.04% high)
For √378 the tangent at 19 wins, missing by only 0.0051. Tangent estimates shine when the number sits close to a perfect square — here 378 is just 17 above 361.
Finding √378 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, 19 (19² = 361):
| Step | Guess x | 378 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 19.8947368421 | 19.4473684211 | 2 |
| 2 | 19.4473684211 | 19.4370771313 | 19.4422227762 | 6 |
| 3 | 19.4422227762 | 19.4422214143 | 19.4422220952 | 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 √378 = 19.4422220952 to every decimal shown.
√378 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √378 the pattern is [19; 2, 3, 1, 4, 1, 3, 2, 38] with the block of 8 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √378 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 |
|---|---|---|
| 19/1 | 19.0000000000 | 4.4 × 10⁻¹ |
| 39/2 | 19.5000000000 | 5.8 × 10⁻² |
| 136/7 | 19.4285714286 | 1.4 × 10⁻² |
| 175/9 | 19.4444444444 | 2.2 × 10⁻³ |
| 836/43 | 19.4418604651 | 3.6 × 10⁻⁴ |
| 1,011/52 | 19.4423076923 | 8.6 × 10⁻⁵ |
The same fractions solve Pell’s equation, x² − 378y² = 1. Its smallest solution in positive whole numbers is x = 8,749, y = 450.
√378 in geometry and everyday measurements
- A square patio or deck of 378 square feet is about 19.44 ft (19 ft 5 in) on each side, so edging all the way around takes 4 × √378 ≈ 77.8 ft.
- 378 is not a sum of two whole-number squares — the prime factor 3 and 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √378 is never the diagonal of a rectangle with whole-number sides. It is, however, the space diagonal of a 1 × 4 × 19 box, because 1² + 4² + 19² = 378.
- Since √378 = 3√42, a length of √378 is exactly 3 copies of the length √42 laid end to end.
Square roots near √378 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √375 | 5√15 | 19.3649 | No |
| √376 | 2√94 | 19.3907 | No |
| √377 | √377 | 19.4165 | No |
| √378 | 3√42 | 19.4422 | No |
| √379 | √379 | 19.4679 | No |
| √380 | 2√95 | 19.4936 | No |
| √381 | √381 | 19.5192 | No |
- The cube root of 378 is about 7.230427.
- Squaring undoes the root: (√378)² = 378, while 378² = 142,884 — the number whose square root is 378.
Frequently asked questions
What is the square root of 378?
The square root of 378 is 3√42 in simplest radical form, which is about 19.4422220952. The negative root, −19.442222, also squares to 378.
Is the square root of 378 rational or irrational?
Irrational. 378 is not a perfect square — it falls between 361 and 400 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √378 be simplified?
Yes. The largest perfect square dividing 378 is 9, so √378 = √9 × √42 = 3√42.
What is √378 rounded to two decimal places?
√378 ≈ 19.44 to two decimal places (19.4 to one, 19.442 to three). Check: 19.44² = 377.9136, close to 378.