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 are the solutions for the simplification problems. a) Simplify 3^n+1 + 9 · 3^n6 · 3^n + 3^n+1 Step 1: Rewrite terms using exponent rules (a^m+n = a^m · a^n) and express 9 as a power of 3. 3^n+1 = 3^n · 3^1 = 3 · 3^n 9 = 3^2 Substitute these into the expression: (3 · 3^n + 3^2 · 3^n)/(6 · 3^n + 3 · 3^n) Step 2: Factor out the common term 3^n from both the numerator and the denominator. (3^n (3 + 3^2))/(3^n (6 + 3)) (3^n (3 + 9))/(3^n (9)) (3^n (12))/(3^n (9)) Step 3: Cancel out 3^n and simplify the fraction. (12)/(9) = (4)/(3) The simplified expression is (4)/(3). b) Simplify 5^-2x+1 - 25^x3 · 5^-2x+1 + 25^-x Step 1: Rewrite all terms with base 5 using exponent rules (a^m+n = a^m · a^n and (a^m)^n = a^mn). 5^-2x+1 = 5^-2x · 5^1 = 5 · 5^-2x 25^x = (5^2)^x = 5^2x 25^-x = (5^2)^-x = 5^-2x Step 2: Substitute these rewritten terms into the expression. Numerator: 5^-2x+1 - 25^x = 5 · 5^-2x - 5^2x Denominator: 3 · 5^-2x+1 + 25^-x = 3 · (5 · 5^-2x) + 5^-2x = 15 · 5^-2x + 5^-2x Factor out 5^-2x from the denominator: = 5^-2x (15 + 1) = 16 · 5^-2x Step 3: Form the simplified fraction. 5 · 5^-2x - 5^2x16 · 5^-2x Step 4: Divide each term in the numerator by the denominator. 5 · 5^-2x16 · 5^-2x - 5^2x16 · 5^-2x Step 5: Simplify each term. The first term: 5 · 5^-2x16 · 5^-2x = (5)/(16) The second term: 5^2x16 · 5^-2x = (1)/(16) · 5^2x5^-2x Using the exponent rule $(a^m)/(a^n) = a