How do I factor 28 to simplify the square root?

Factoring 28 to simplify its square root involves breaking down the number into its prime factors and then identifying which factors form perfect squares that can be extracted from under the radical sign. This systematic approach is fundamental to simplifying radicals and is an essential technique in Class 10 algebra. Understanding the factorization process not only helps with square roots but also strengthens your overall number sense and algebraic manipulation skills.

The factorization process for 28 can be approached in several ways, but the most systematic method is to find the complete prime factorization. Starting with 28, we divide by the smallest prime number that goes into it evenly. Since 28 is even, we know 2 is a factor: 28 ÷ 2 = 14. Continuing with 14, which is also even: 14 ÷ 2 = 7. Now we have 7, which is a prime number, so we stop. This gives us the complete prime factorization: 28 = 2 × 2 × 7, or written with exponents: 28 = 2² × 7. This factorization is crucial because it reveals that 28 contains the perfect square 2² (which equals 4). Since √(2²) = 2, we can extract this factor from under the radical sign.

Applying this factorization to simplify √28: We write √28 = √(2² × 7). Using the property that the square root of a product equals the product of the square roots, we get √(2² × 7) = √(2²) × √7 = 2√7. The key insight is recognizing that whenever you have a pair of identical prime factors under a radical, you can extract one of those factors from the pair outside the radical. In this case, the pair of 2s becomes a single 2 outside the radical, while the unpaired 7 remains inside. If we had encountered a factor that appeared three times, such as in √(2³ × 7) = √(2² × 2 × 7), we would extract one 2 from the pair (2²) and leave one 2 under the radical: 2√(2 × 7) = 2√14.