Question

solve: \\(3125^{4x-4} = -125\\)

one solution:
no solution

write answers as integers or reduced fractions.

Explanation:

Analyze the given equation

We are given the exponential equation:

$$3125^{4x-4} = -125$$

Examine the base and exponent

The base on the left-hand side is \(3125\), which is a positive real number (\(3125 = 5^5\)).
For any real exponent \(y\), a positive base raised to any power is always strictly positive:

$$3125^{4x-4} > 0$$

Compare with the right-hand side

The right-hand side of the equation is \(-125\), which is a negative number.
Since a positive number can never equal a negative number, we have:

$$3125^{4x-4} eq -125$$

Determine the solution set

Because the left-hand side is strictly positive for all real values of \(x\) and the right-hand side is negative, there is no real value of \(x\) that satisfies this equation.
Thus, the equation has no solution.

Answer:

• One solution:
• No solution (Correct answer)