1.1.3 \( 125^{3 x-2}=25^{4 x+10} \)

To solve the equation \( 125^{3x-2} = 25^{4x+10} \), we can express both sides with a common base. Notably, \( 125 = 5^3 \) and \( 25 = 5^2 \). First, rewrite the equation: \[ (5^3)^{3x-2} = (5^2)^{4x+10} \] Using the power of a power property \((a^m)^n = a^{mn}\), we get: \[ 5^{3(3x-2)} = 5^{2(4x+10)} \] This simplifies to: \[ 5^{9x - 6} = 5^{8x + 20} \] Because the bases are the same, we can set the exponents equal to one another: \[ 9x - 6 = 8x + 20 \] Now, isolate \(x\): \[ 9x - 8x = 20 + 6 \] \[ x = 26 \] Thus, the solution is \( x = 26 \).