Which is a perfect square, 28 or 32?

Neither 28 nor 32 is a perfect square. This can be verified quickly by noting that both numbers fall between the consecutive perfect squares 25 (which is 5²) and 36 (which is 6²), meaning neither can be the square of a whole number. Understanding how to identify perfect squares is a fundamental skill in Class 10 mathematics that appears across multiple topics, from simplifying radicals to solving quadratic equations to working with the Pythagorean theorem.

To understand why neither 28 nor 32 is a perfect square, let's examine their prime factorizations. For 28: 28 = 4 × 7 = 2² × 7. While the factor 2 appears twice (forming a pair), the factor 7 appears only once. For a number to be a perfect square, all prime factors must appear an even number of times, forming complete pairs. The unpaired 7 prevents 28 from being a perfect square. For 32: 32 = 2⁵ = 2 × 2 × 2 × 2 × 2. Here, the prime factor 2 appears five times, which is odd. Despite having only one distinct prime factor, the odd exponent disqualifies 32 from being a perfect square. If 32 had one more factor of 2 (making 2⁶ = 64), it would be a perfect square (8²), but as it stands, with 2⁵, it is not.

We can also verify this by calculating their square roots: √28 ≈ 5.29 and √32 ≈ 5.66. Neither is a whole number, confirming that neither 28 nor 32 is a perfect square. In simplified radical form, √28 = 2√7 and √32 = 4√2, which further confirms their non-perfect-square status since both simplified forms still contain radicals. If either had been a perfect square, the simplification would have resulted in a whole number with no radical remaining.