JAMB Mathematics 2015 Paper

Concept:Write both sides of the exponential equation with the same base, then equate their exponents.Explanation:Rewrite $\frac{1}{125}$ as $125^{-1}$.Since $125 = 5^3$, we get $\frac{1}{125} = (5^3)^{-1} = 5^{-3}$.Substitute into the right-hand side:$$\left(\frac{1}{125}\right)^{x+3} = (5^{-3})^{x+3}$$Using the power rule $(a^m)^n = a^{mn}$, simplify:$$(5^{-3})^{x+3} = 5^{-3(x+3)} = 5^{-3x-9}$$The original equation becomes:$$5^{x-2} = 5^{-3x-9}$$Since the bases are equal, the exponents must be equal:$$x - 2 = -3x - 9$$Collect like terms:$$x + 3x = -9 + 2$$$$4x = -7$$Divide both sides by $4$:$$x = \frac{-7}{4}$$Answer:$$x = -\frac{7}{4}$$This corresponds to option B.