Final answer:
To find the negative value of m when n=8, we use the given relation m² is directly proportional to n³ and initial condition m=8 when n=2 to find the constant of proportionality. Using this constant, we determine that m=-64 for n=8.
Step-by-step explanation:
The question asks to find the negative value of m when n=8, given that m² is directly proportional to n³ and m=8 when n=2. First, we establish the proportionality by stating that m² = k * n³, where k is the constant of proportionality. Plugging in the given values, we calculate k by solving 8² = k * 2³, which gives us k = 16. With k in hand, we can now find m for n=8 by solving the equation m² = 16 * 8³. This results in m² = 8192, and taking the square root gives us m = √8192. Since we want the negative value, we take the negative root to get m = -√8192, which simplifies to m = -64.