222 is not a perfect square because its prime factorization contains prime factors that are not raised to an even power.
Understanding Perfect Squares
A perfect square is an integer that can be obtained by squaring another integer. For instance, 9 is a perfect square because it equals 32. The prime factorization of a perfect square will always have even exponents for all its prime factors.
Prime Factorization of 222
According to the provided reference, the prime factorization of 222 is 21 × 31 × 371.
- 2 appears with an exponent of 1.
- 3 appears with an exponent of 1.
- 37 appears with an exponent of 1.
Why 222 Fails as a Perfect Square
In the prime factorization of 222, none of the prime factors (2, 3, and 37) have even exponents. To be a perfect square, all prime factors must have even exponents. As the reference states, "the prime factor 2 is not in the pair," meaning its exponent is not an even number (like 2, 4, 6, etc.). Therefore, 222 is not a perfect square.
