h) \( \frac{1}{25^{x}}=125 \)

To solve the equation \( \frac{1}{25^{x}}=125 \), we can rewrite both sides in terms of powers of 5. Since \( 25 = 5^2 \) and \( 125 = 5^3 \), we get: \[ \frac{1}{(5^2)^{x}} = 5^3 \] This simplifies to: \[ 5^{-2x} = 5^3 \] Since the bases are the same, we can set the exponents equal to each other: \[ -2x = 3 \] Solving for \( x \), we divide both sides by -2: \[ x = -\frac{3}{2} \] Now we can write the solution in a more engaging way: feel free to think of exponents as little armies! When you set them equal, it's just like saying two groups of warriors must have the same strength. So, when the power squad from the left side (with a negative twist) meets the mighty 3 from the right, they balance like an epic standoff in a fantasy tale!