What is √28 in simplest radical form?

The simplest radical form of √28 is 2√7. This representation is considered "simplest" because all perfect square factors have been extracted from under the radical sign, leaving only a prime number (7) that cannot be simplified further. Understanding what constitutes simplest radical form is crucial for Class 10 mathematics, as this form is often required in examinations and provides the most elegant way to express irrational square roots.

A radical expression is in simplest form when three conditions are met: first, no perfect square factors other than 1 remain under the radical; second, no fractions appear under the radical; and third, no radicals appear in the denominator of a fraction (this is called rationalizing the denominator, though it applies to different types of problems). For √28, we need only concern ourselves with the first condition. To achieve simplest radical form, we factor 28 as 4 × 7, recognize that 4 is a perfect square (2²), and extract it: √28 = √(4 × 7) = √4 × √7 = 2 × √7 = 2√7. The number 7 remaining under the radical is prime, meaning it has no factors other than 1 and itself, so no further simplification is possible.

The importance of expressing radicals in simplest form extends beyond mere convention—it serves practical purposes in mathematical work. Simplified radicals are easier to compare (for example, it's clearer that 2√7 is greater than 3√2 when you calculate approximate values), simpler to combine in addition and subtraction (you can only combine like radicals, just as you combine like terms in algebra), and more straightforward to use in multiplication and division. Consider the expression √28 × √63: if you work with the original forms, the calculation is cumbersome, but if you first simplify to 2√7 × 3√7 = 6 × 7 = 42, the answer becomes immediately clear. Additionally, presenting answers in simplest radical form demonstrates mathematical proficiency and attention to detail, both valued in academic assessments.