√28 at a glance
- Exact value
- 2√7
- Decimal (10 places)
- 5.2915026221
- Rounded
- 5.3 · 5.29 · 5.292
- Perfect square?
- No — between 5² and 6²
- Rational?
- Irrational
- Both square roots
- ±5.291503
- Prime factorization
- 2² × 7
- Cube root
- 3.036589
How to simplify √28
Look for the largest perfect square that divides 28. Here it is 4 (2²), because 28 = 4 × 7 and 7 has no square factor left:
The prime factorization tells the same story: 28 = 2² × 7. Each pair of equal primes leaves the radical as one factor, so 2 comes out and 7 stays inside.
Check: (2√7)² = 2² × 7 = 4 × 7 = 28. As a decimal, 2√7 = 2 × 2.6457513111 ≈ 5.2915026221.
Where √28 sits between perfect squares
25 = 5² and 36 = 6² are the nearest perfect squares, so √28 lies between 5 and 6. 28 is 3 above 25 and 8 below 36, so the root is closer to 5.
- Straight line between 25 and 36: 5.2727 (0.35% low)
- Tangent from 5, i.e. 5 + 3 ÷ 10: 5.3000 (0.16% high)
- Tangent from 6, i.e. 6 − 8 ÷ 12: 5.3333 (0.79% high)
For √28 the tangent at 5 wins, missing by only 0.0085. Tangent estimates shine when the number sits close to a perfect square — here 28 is just 3 above 25.
Finding √28 with the Babylonian method
Picture a rectangle with an area of 28 and one side x; the other side must be 28 ÷ x. Averaging the two sides gives a rectangle that is closer to a square, and the side of that square is exactly √28.
Start from the nearest whole number, 5 (5² = 25):
| Step | Guess x | 28 ÷ x | Average | Correct decimals |
|---|---|---|---|---|
| 1 | 5.0000000000 | 5.6000000000 | 5.3000000000 | 2 |
| 2 | 5.3000000000 | 5.2830188679 | 5.2915094340 | 5 |
| 3 | 5.2915094340 | 5.2914958103 | 5.2915026221 | 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 √28 = 5.2915026221 to every decimal shown.
√28 as a continued fraction
Every irrational square root has a continued fraction that repeats. For √28 the pattern is [5; 3, 2, 3, 10] with the block of 4 terms after the semicolon repeating forever. A pattern that never ends is one more proof that √28 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 |
|---|---|---|
| 5/1 | 5.0000000000 | 2.9 × 10⁻¹ |
| 16/3 | 5.3333333333 | 4.2 × 10⁻² |
| 37/7 | 5.2857142857 | 5.8 × 10⁻³ |
| 127/24 | 5.2916666667 | 1.6 × 10⁻⁴ |
| 1,307/247 | 5.2914979757 | 4.6 × 10⁻⁶ |
| 4,048/765 | 5.2915032680 | 6.5 × 10⁻⁷ |
The same fractions solve Pell’s equation, x² − 28y² = 1. Its smallest solution in positive whole numbers is x = 127, y = 24.
√28 in geometry and everyday measurements
- A square tile with an area of 28 square inches has sides about 5.292 in long.
- 28 is not a sum of two whole-number squares — the prime factor 7 (one less than a multiple of 4) appears an odd number of times, which rules that out — so √28 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 √28 as its space diagonal.
- Since √28 = 2√7, a length of √28 is exactly 2 copies of the length √7 laid end to end.
Square roots near √28 and related roots
| Root | Simplest form | Decimal | Perfect square? |
|---|---|---|---|
| √25 | 5 | 5.0000 | Yes |
| √26 | √26 | 5.0990 | No |
| √27 | 3√3 | 5.1962 | No |
| √28 | 2√7 | 5.2915 | No |
| √29 | √29 | 5.3852 | No |
| √30 | √30 | 5.4772 | No |
| √31 | √31 | 5.5678 | No |
- The cube root of 28 is about 3.036589.
- Four times the radicand doubles the root: √112 = 2 × √28 ≈ 10.583005.
Frequently asked questions
What is the square root of 28?
The square root of 28 is 2√7 in simplest radical form, which is about 5.2915026221. The negative root, −5.291503, also squares to 28.
Is the square root of 28 rational or irrational?
Irrational. 28 is not a perfect square — it falls between 25 and 36 — and the square root of any whole number that is not a perfect square cannot be written as a fraction.
Can √28 be simplified?
Yes. The largest perfect square dividing 28 is 4, so √28 = √4 × √7 = 2√7.
What is √28 rounded to two decimal places?
√28 ≈ 5.29 to two decimal places (5.3 to one, 5.292 to three). Check: 5.29² = 27.9841, close to 28.