Final answer:
To determine the values of n that make the expression negative, we need to examine the signs of each factor and determine whether the exponents are odd or even. If the exponent is odd, n must be an integer that satisfies 3n = 2k + 1. If the exponent is even, n must be an integer that satisfies 3n = 2k.
Step-by-step explanation:
To determine for what natural numbers n the expression (-27.1)^3n * (7.1)^5n is negative, we need to examine the signs of each factor. A negative number raised to an even power gives a positive result, while a negative number raised to an odd power gives a negative result. Since (-27.1) is negative and (7.1) is positive, we need to find the values of n that make the exponents either odd or even.
If the exponent (3n) is odd, then n must be an integer that satisfies 3n = 2k + 1, where k is an integer. If the exponent (3n) is even, then n must be an integer that satisfies 3n = 2k, where k is an integer.
For example, if n = 1, the exponents become 3(1) = 3 and 5(1) = 5, which are both odd. Therefore, the expression (-27.1)^3n * (7.1)^5n will be negative when n = 1. Similarly, if n = 2, the exponents become 3(2) = 6 and 5(2) = 10, which are both even. Therefore, the expression (-27.1)^3n * (7.1)^5n will also be negative when n = 2.