√368 at a glance
- Exact value
- 4√23
- Decimal (10 places)
- 19.1833260933
- Rounded
- 19.2 · 19.18 · 19.183
- Perfect square?
- No — between 19² and 20²
- Rational?
- Irrational
- Both square roots
- ±19.183326
- Prime factorization
- 2⁴ × 23
- Cube root
- 7.166096
How to simplify √368
Look for the largest perfect square that divides 368. Here it is 16 (4²), because 368 = 16 × 23 and 23 has no square factor left:
The prime factorization tells the same story: 368 = 2⁴ × 23. Each pair of equal primes leaves the radical as one factor, so 2² comes out and 23 stays inside.
368 has 2 square factors (4 and 16). Starting with a smaller one still works but takes more rounds: √368 = 2√92, and √92 can be simplified again. Using 16 straight away finishes in one step.
Check: (4√23)² = 4² × 23 = 16 × 23 = 368. As a decimal, 4√23 = 4 × 4.7958315233 ≈ 19.1833260933.
Where √368 sits between perfect squares
361 = 19² and 400 = 20² are the nearest perfect squares, so √368 lies between 19 and 20. 368 is 7 above 361 and 32 below 400, so the root is closer to 19.
- Straight line between 361 and 400: 19.1795 (0.02% low)
- Tangent from 19, i.e. 19 + 7 ÷ 38: 19.1842 (0% high)
- Tangent from 20, i.e. 20 − 32 ÷ 40: 19.2000 (0.09% high)
For √368 the tangent at 19 wins, missing by only 0.0009. Tangent estimates shine when the number sits close to a perfect square — here 368 is just 7 above 361.
Finding √368 with the Babylonian method
Picture a rectangle with an area of 368 and one side x; the other side must be 368 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √368.
Start from the nearest whole number, 19 (19² = 361):
| Step | Guess x | 368 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 19.0000000000 | 19.3684210526 | 19.1842105263 | 3 |
| 2 | 19.1842105263 | 19.1824417010 | 19.1833261136 | 7 |
| 3 | 19.1833261136 | 19.1833260729 | 19.1833260933 | 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 √368 = 19.1833260933 to every decimal shown.
√368 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √368 the pattern is [19; 5, 2, 5, 38] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √368 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 | 1.8 × 10⁻¹ |
| 96/5 | 19.2000000000 | 1.7 × 10⁻² |
| 211/11 | 19.1818181818 | 1.5 × 10⁻³ |
| 1,151/60 | 19.1833333333 | 7.2 × 10⁻⁶ |
| 43,949/2,291 | 19.1833260585 | 3.5 × 10⁻⁸ |
| 220,896/11,515 | 19.1833260964 | 3.1 × 10⁻⁹ |
The same fractions solve Pell’s equation, x² − 368y² = 1. Its smallest solution in positive whole numbers is x = 1,151, y = 60.
√368 in geometry and everyday measurements
- A square patio or deck of 368 square feet is about 19.18 ft (19 ft 2 in) on each side, so edging all the way around takes 4 × √368 ≈ 76.7 ft.
- 368 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 √368 is never the diagonal of a rectangle with whole-number sides. It is not a sum of three positive squares either, so no whole-number box has √368 as its space diagonal.
- Since √368 = 4√23, a length of √368 is exactly 4 copies of the length √23 laid end to end.
Square roots near √368 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √365 | √365 | 19.1050 | No |
| √366 | √366 | 19.1311 | No |
| √367 | √367 | 19.1572 | No |
| √368 | 4√23 | 19.1833 | No |
| √369 | 3√41 | 19.2094 | No |
| √370 | √370 | 19.2354 | No |
| √371 | √371 | 19.2614 | No |
- The cube root of 368 is about 7.166096.
- Because 368 = 4 × 92, the root is twice √92: 2 × 9.591663 ≈ 19.183326.
Frequently asked questions
What is the square root of 368?
The square root of 368 is 4√23 in simplest radical form, which is about 19.1833260933. The negative root, −19.183326, also squares to 368.
Is the square root of 368 rational or irrational?
Irrational. 368 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 √368 be simplified?
Yes. The largest perfect square dividing 368 is 16, so √368 = √16 × √23 = 4√23.
What is √368 rounded to two decimal places?
√368 ≈ 19.18 to two decimal places (19.2 to one, 19.183 to three). Check: 19.18² = 367.8724, close to 368.