This mathematics problem involves applying core mathematical principles and formulas. Below you will find a complete step-by-step solution with detailed explanations for each step, helping you understand not just the answer but the method behind it.
ScanSolve AI Answer
Here's how to simplify the expression: Step 1: Identify common factors in the numerator. The numerator is 5^n - 5^n-2. We can factor out 5^n-2. Recall that 5^n = 5^n-2 · 5^2. 5^n - 5^n-2 = 5^n-2 · 5^2 - 5^n-2 · 1 = 5^n-2(5^2 - 1) 5^n-2(25 - 1) = 5^n-2(24) Step 2: Identify common factors in the denominator. The denominator is 5^3 × 5^n - 125 × 5^n-2. First, calculate 5^3: 5^3 = 5 × 5 × 5 = 125. Substitute this value into the denominator: 125 × 5^n - 125 × 5^n-2 Now, factor out 125: 125(5^n - 5^n-2) From Step 1, we know that 5^n - 5^n-2 = 5^n-2(24). Substitute this into the denominator: 125 · 5^n-2(24) Step 3: Substitute the simplified numerator and denominator back into the fraction. 5^n-2(24)125 · 5^n-2(24) Step 4: Cancel out common terms from the numerator and denominator. Both the numerator and denominator have the term 5^n-2(24). 5^n-2(24)125 · 5^n-2(24) = (1)/(125) The simplified expression is: (1)/(125)