When a square number is written as a product of prime numbers, the factors will have a special property. Each prime factor will be raised to an even exponent. This is because when a number is squared, each of its prime factors is doubled in exponent. For example, if we have a square number like 36, which can be expressed as 2² × 3², both the prime factors 2 and 3 are raised to an even exponent (2). However, it is important to note that this property is specific to square numbers and may not apply to other types of numbers.
When a square number is written as a product of prime numbers, we can make the following observations about its factors:
1. Every prime factor appears with an even exponent:
Since the number is a perfect square, each prime factor must be multiplied by itself to yield a square. Therefore, all prime factors will have an even exponent in their representation.
2. The factors are unique:
When a square number is factored into primes, the prime factors are unique. This uniqueness is a consequence of the Fundamental Theorem of Arithmetic, which states that every positive integer can be expressed uniquely as a product of primes (up to the order of factors).
3. The factors can be rearranged:
When expressing a square number as a product of prime factors, the order in which the factors are written does not matter. This property follows from the commutative property of multiplication.
For example, let's consider the square number 36. It can be expressed as a product of prime factors as 2^2 * 3^2. Here, both 2 and 3 appear with even exponents, and the factors are unique. We could also write it as 3^2 * 2^2, and the result would still be the same.
Overall, when a square number is factored into primes, the factors have even exponents, are unique, and can be rearranged without changing the result.
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Step-by-step explanation:
When a square number is written as a product of prime numbers, the factors will have a special property. Each prime factor will be raised to an even exponent. This is because when a number is squared, each of its prime factors is doubled in exponent. For example, if we have a square number like 36, which can be expressed as 2² × 3², both the prime factors 2 and 3 are raised to an even exponent (2). However, it is important to note that this property is specific to square numbers and may not apply to other types of numbers.
Answer:
When a square number is written as a product of prime numbers, we can make the following observations about its factors:
1. Every prime factor appears with an even exponent:
2. The factors are unique:
3. The factors can be rearranged:
Overall, when a square number is factored into primes, the factors have even exponents, are unique, and can be rearranged without changing the result.