Simplifying the square root of 28 means rewriting √28 as 2√7 by factoring out perfect squares. This reduces the expression to its simplest radical form, making it easier to use in equations and comparisons.
The square root of 28 is irrational-it cannot be written as a fraction-but it can be expressed in simpler radical form. The process uses prime factorization to identify perfect square factors within 28. Since 28 = 4 × 7 and 4 is a perfect square (2²), √28 becomes √(4×7) = √4 × √7 = 2√7.
This simplification is essential in algebra, trigonometry, and geometry. For example, when solving quadratic equations or calculating distances using the Pythagorean theorem, simplified radicals reduce calculation errors and improve clarity. The result, 2√7, is approximately 5.29, but keeping it as 2√7 preserves precision.
Simplifying radicals relies on the property √(a × b) = √a × √b, valid for non-negative a and b. To apply this to √28, first find all factor pairs of 28: (1, 28), (2, 14), (4, 7). Among these, only 4 is a perfect square.
Break 28 into 4 × 7. Then apply the rule: √(4 × 7) = √4 × √7. Since √4 = 2, the expression becomes 2√7. No further simplification is possible because 7 is prime and has no square factors other than 1.
Using a factor tree confirms this: 28 splits into 2 × 14, then 2 × 2 × 7. The two 2s form 2², which is a perfect square. So √28 = √(2² × 7) = 2√7. This method applies to any integer under a square root.
Radicals fall into three categories based on their structure:
| Type | Definition | Example | Can Be Simplified? |
|---|---|---|---|
| Perfect square radicand | Number under root is a perfect square | √16 = 4 | Yes, fully |
| Non-perfect square with square factor | Contains a perfect square factor greater than 1 | √28 = 2√7 | Yes, partially |
| Prime radicand | Number under root is prime | √17 | No |
√28 belongs to the second category. It contains a perfect square factor (4), so it simplifies. But √7 cannot be broken down further because 7 is prime and has no square divisors except 1.
A common mistake is assuming √28 = √4 × √8. That's incorrect because 4 × 8 = 32, not 28. Always verify factor pairs multiply to the original number. Another error is stopping at √(2×14) without checking whether either factor is a perfect square.
Simplified radicals are required in standardized tests like the SAT and ACT. Questions often ask for "simplest radical form," and answers like √28 lose points compared to 2√7. According to the College Board's official practice materials, responses must be fully simplified to earn full credit.
In physics, simplified radicals appear in formulas involving velocity, acceleration, and wave motion. For example, the time it takes for an object to fall from height h is t = √(2h/g). If h = 28 meters and g ≈ 9.8 m/s², then t = √(56/9.8) ≈ √5.71. But if the problem involves √28 directly, simplifying it early avoids rounding errors.
I once worked with a high school student who struggled with a geometry problem involving a right triangle with legs of length 2 and √28. He kept entering √28 into his calculator and getting decimal approximations. After showing him how to rewrite √28 as 2√7, he realized the hypotenuse was √(4 + 4×7) = √32 = 4√2. The pattern became clear only after simplification.
Many people think simplifying means reducing the number under the root to the smallest possible value. But simplification means factoring out perfect squares-not minimizing the radicand. √28 is not simplified as √7; it must include the coefficient 2.
Another misconception is that √28 and √7 are equivalent. They are not. √28 ≈ 5.29, while √7 ≈ 2.65. The difference is exactly a factor of 2. This shows why the coefficient matters.
Some assume all roots can be simplified. But √13, √17, and √19 cannot be simplified because they are prime numbers. The only way to simplify √28 is through its factor 4, which is unique to composite numbers with square factors.
Step 1: Factor 28 into primes: 28 = 2 × 2 × 7 = 2² × 7.
Step 2: Apply the rule √(a² × b) = a√b: √(2² × 7) = 2√7.
Step 3: Check if √7 can be simplified. 7 is prime, so it cannot. Final answer: 2√7.
Verification: Square 2√7: (2√7)² = 4 × 7 = 28. This confirms correctness.
When solving equations like x² = 28, the solution is x = ±√28 = ±2√7. Writing it this way avoids confusion and aligns with standard mathematical notation.