If 25 1 - x times 5 x 2 div frac 1 125 x 625 -1... - JAMB Mathematics 2024 Question

We are given the equation: \(25^{1 - x} \times 5^{x + 2} \div (\frac{1}{125})^{x} = 625^{-1}\). Start by expressing all numbers in terms of base 5:

• \(25 = 5^2\), so \(25^{1 - x} = (5^2)^{1 - x} = 5^{2(1 - x)} = 5^{2 - 2x}\).
• \(125 = 5^3\), so \((\frac{1}{125})^x = (5^{-3})^x = 5^{-3x}\).
• \(625 = 5^4\), so \(625^{-1} = (5^4)^{-1} = 5^{-4}\).

Now, the equation becomes: \(5^{2 - 2x} \times 5^{x + 2} \div 5^{-3x} = 5^{-4}\).

Using the laws of exponents, combine the terms: \(5^{2 - 2x} \times 5^{x + 2} \div 5^{-3x} = 5^{2 - 2x + x + 2 + 3x} = 5^{4 + 2x}\).

Set the exponents equal: \(4 + 2x = -4\), so solving for x gives: \(2x = -8\), hence \(x = -4\).