√8 at a glance
- Exact value
- 2√2
- Decimal (10 places)
- 2.8284271247
- Rounded
- 2.8 · 2.83 · 2.828
- Perfect square?
- No — between 2² and 3²
- Rational?
- Irrational
- Both square roots
- ±2.828427
- Prime factorization
- 2³
- Cube root
- 2
How to simplify √8
Look for the largest perfect square that divides 8. Here it is 4 (2²), because 8 = 4 × 2 and 2 has no square factor left:
The prime factorization tells the same story: 8 = 2³. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 2 stays inside.
Check: (2√2)² = 2² × 2 = 4 × 2 = 8. As a decimal, 2√2 = 2 × 1.4142135624 ≈ 2.8284271247.
Where √8 sits between perfect squares
4 = 2² and 9 = 3² are the nearest perfect squares, so √8 lies between 2 and 3. 8 is 4 above 4 and 1 below 9, so the root is closer to 3.
- Straight line between 4 and 9: 2.8000 (1.01% low)
- Tangent from 2, i.e. 2 + 4 ÷ 4: 3.0000 (6.07% high)
- Tangent from 3, i.e. 3 − 1 ÷ 6: 2.8333 (0.17% high)
For √8 the tangent at 3 wins, missing by only 0.0049. Tangent estimates shine when the number sits close to a perfect square — here 8 is just 1 below 9.
Finding √8 with the Babylonian method
Picture a rectangle with an area of 8 and one side x; the other side must be 8 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √8.
Start from the nearest whole number, 3 (3² = 9):
| Step | Guess x | 8 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 3.0000000000 | 2.6666666667 | 2.8333333333 | 2 |
| 2 | 2.8333333333 | 2.8235294118 | 2.8284313725 | 5 |
| 3 | 2.8284313725 | 2.8284228769 | 2.8284271247 | 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 √8 = 2.8284271247 to every decimal shown.
√8 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √8 the pattern is [2; 1, 4] with the block of 2 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √8 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 |
|---|---|---|
| 2/1 | 2.0000000000 | 8.3 × 10⁻¹ |
| 3/1 | 3.0000000000 | 1.7 × 10⁻¹ |
| 14/5 | 2.8000000000 | 2.8 × 10⁻² |
| 17/6 | 2.8333333333 | 4.9 × 10⁻³ |
| 82/29 | 2.8275862069 | 8.4 × 10⁻⁴ |
| 99/35 | 2.8285714286 | 1.4 × 10⁻⁴ |
The same fractions solve Pell’s equation, x² − 8y² = 1. Its smallest solution in positive whole numbers is x = 3, y = 1.
√8 in geometry and everyday measurements
- A square tile with an area of 8 square inches has sides about 2.828 in long.
- 8 = 2² + 2², so by the Pythagorean theorem √8 is the diagonal of a 2 × 2 rectangle — and the distance between the points (0, 0) and (2, 2) on a grid.
- Since √8 = 2√2, a length of √8 is exactly 2 copies of the length √2 laid end to end.
Square roots near √8 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √5 | √5 | 2.2361 | No |
| √6 | √6 | 2.4495 | No |
| √7 | √7 | 2.6458 | No |
| √8 | 2√2 | 2.8284 | No |
| √9 | 3 | 3.0000 | Yes |
| √10 | √10 | 3.1623 | No |
| √11 | √11 | 3.3166 | No |
- The cube root of 8 is exactly 2 — 8 is a perfect cube as well (2³).
- Multiplying the radicand by 100 multiplies the root by 10: √800 = 10 × √8 ≈ 28.284271.
Frequently asked questions
What is the square root of 8?
The square root of 8 is 2√2 in simplest radical form, which is about 2.8284271247. The negative root, −2.828427, also squares to 8.
Is the square root of 8 rational or irrational?
Irrational. 8 is not a perfect square — it falls between 4 and 9 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √8 be simplified?
Yes. The largest perfect square dividing 8 is 4, so √8 = √4 × √2 = 2√2.
What is √8 rounded to two decimal places?
√8 ≈ 2.83 to two decimal places (2.8 to one, 2.828 to three). Check: 2.83² = 8.0089, close to 8.