Final answer:
To simplify 5√x, identify if x has any perfect square factors and simplify accordingly. If x is already a perfect square, it can be simplified out of the square root; otherwise, 5√x is already in its simplest form.
Step-by-step explanation:
The question asks us to simplify 5√x using prime factors. To do this, we need to understand a few key concepts about exponents and roots, in particular how they relate to prime factorization and simplifying square roots. Prime factors are the building blocks of numbers, they are the primes that multiply together to give the original number. In terms of simplifying square roots using prime factors, if we have a perfect square under the root, it can be taken out of the root as a single number.
According to our given reference, we can consider the expression 5√x as a variant of x² = √x, which implies that the square root of a number is equivalent to that number raised to the power of 1/2. So, 5√x can be re-expressed as 5x⅛. However, simplifying 5√x doesn't require further action, since it is already in its simplest form unless we know the prime factors of x that could potentially be squared and taken out of the square root.
For example, if x is a number such as 25, which has a prime factorization of 5², then the expression becomes 5√(5²), which simplifies to 5×5 = 25. Conversely, if x is a number like 2, which does not have a square number in its prime factorization, the expression remains 5√2.