Why is 28 not a perfect square?

The number 28 is not a perfect square because there is no whole number that, when multiplied by itself, equals 28. A perfect square is defined as the product of an integer with itself—for example, 25 = 5 × 5, 36 = 6 × 6, and 49 = 7 × 7. Since √28 ≈ 5.2915..., which is not a whole number, we can conclusively state that 28 is not a perfect square. This concept is fundamental to understanding number properties in Class 10 mathematics and has important implications for working with radicals, solving equations, and understanding number theory.

To understand why 28 fails to be a perfect square at a deeper level, we need to examine its prime factorization: 28 = 2² × 7. Here's the critical insight: for a number to be a perfect square, every prime factor in its prime factorization must appear an even number of times (in pairs). Looking at 28's factorization, the prime factor 2 appears twice (2²), which is even and forms a complete pair. However, the prime factor 7 appears only once, which is odd. This single, unpaired factor of 7 prevents 28 from being a perfect square. If 28 had been 2² × 7² = 4 × 49 = 196 instead, it would be a perfect square (in fact, 196 = 14²). The presence of that unpaired 7 is precisely what makes 28 a non-perfect square and what causes √28 to be irrational.

This prime factorization rule provides a systematic way to determine whether any number is a perfect square without having to calculate its square root. Consider these examples: Is 144 a perfect square? Factor it: 144 = 2⁴ × 3² = 2 × 2 × 2 × 2 × 3 × 3. The factor 2 appears four times (even), and the factor 3 appears twice (even). Since all prime factors appear an even number of times, 144 is indeed a perfect square (12²). Is 72 a perfect square? Factor it: 72 = 2³ × 3² = 2 × 2 × 2 × 3 × 3. The factor 2 appears three times (odd), while 3 appears twice (even). Since the exponent of 2 is odd, 72 is not a perfect square, and √72 = 6√2, which is irrational.