How do you simplify √28?

Simplifying √28 involves a systematic process of factoring the number 28 to identify and extract perfect square factors from under the radical sign. The goal is to express the square root in its simplest radical form, where no perfect square factors (other than 1) remain under the radical. This process is a fundamental skill in Class 10 algebra and demonstrates understanding of both factorization and the properties of radicals.

The step-by-step process for simplifying √28 begins with finding the prime factorization of 28. Breaking down 28: 28 = 2 × 14 = 2 × 2 × 7 = 2² × 7. This factorization reveals that 28 contains the perfect square factor 4 (which is 2²). Using the property that √(a × b) = √a × √b, we can separate the perfect square from the remaining factor: √28 = √(4 × 7) = √4 × √7. Since √4 = 2, we get our final simplified form: √28 = 2√7. This is considered fully simplified because 7 is a prime number with no perfect square factors other than 1.

To verify that 2√7 is indeed the simplest form, we need to confirm that no further simplification is possible. Looking at the number under the radical (7), we check whether it has any perfect square factors. Since 7 is prime, its only factors are 1 and 7, neither of which (except for the trivial factor 1) is a perfect square. Therefore, 2√7 cannot be simplified further. This simplified form offers several advantages in mathematical work: it's more compact than √28, it clearly shows the coefficient (2) separated from the radical part (√7), and it makes subsequent calculations easier. For instance, if you need to add √28 + √7, recognizing that √28 = 2√7 allows you to write this as 2√7 + √7 = 3√7, combining like terms just as you would with variables in algebra.