Final answer:
The correct answer is option (d) u = 0, v = 0 because 0 raised to any power and then multiplied by 0 will always equal 0, satisfying the equation u^1v = u^2v.
Step-by-step explanation:
The student's question is asking which real numbers u and v satisfy the property u1v = u2v. In interpreting the exponent rules and properties of real numbers, we see that the only way for u raised to any power to be equal to u raised to a different power while multiplied by the same number v, the numbers must either be zero or one.
By examining the options given, we can arrive at the correct option in the final answer:
• If u = 1, u1v and u2v will always be equal regardless of v, because 1 raised to any power is 1.
• If v = 0, u1v and u2v will always be 0 regardless of u, because any number multiplied by 0 is 0.
Therefore, the combinations that satisfy the condition are (u = 1, v = 0) and (u = 0, v = 0). Option (d) u = 0, v = 0 is the correct choice. If we plug this into the equation u1v = u2v, we find that 0 = 0, which is true.