Tell whether the statement is Always, Sometimes, or Never True. If the prime factorization of a number has all even exponents, then the number is a perfect square.

Answer: Always True

Step-by-step solution

Step 1: Define prime factorization with even exponents

Let $N$ be a natural number greater than $1$ . If all exponents in the prime factorization of $N$ are even, each exponent can be expressed in the form $2k$ , where $k$ is a positive integer.

Step 2: Express the number as a square

Using the laws of exponents, $(a^k)^2 = a^{2k}$ . Therefore, we can factor out a power of $2$ from every term, grouping the product into the form $M^2$ , where $M = p_1^{k_1} \cdot p_2^{k_2} \cdots p_m^{k_m}$ is an integer.

Step 3: Conclude whether the statement is Always, Sometimes, or Never True

Since $M$ is a product of integers, $M$ is an integer. Thus, $N$ is the square of an integer $M$ , which is the exact definition of a perfect square. This property holds for every such number without exception.

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