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 the solution to problem 1. a) Simplify 6^3n+1 · 9^3n3^n+1 Step 1: Express the bases in terms of their prime factors. 6 = 2 × 3 9 = 3^2 Step 2: Substitute the prime factors into the expression. (2 × 3)^3n+1 · (3^2)^3n3^n+1 Step 3: Apply the exponent rules (ab)^m = a^m b^m and (a^m)^n = a^mn. 2^3n+1 · 3^3n+1 · 3^2 · 3n3^n+1 2^3n+1 · 3^3n+1 · 3^6n3^n+1 Step 4: Combine terms with the same base in the numerator using a^m · a^n = a^m+n. 2^3n+1 · 3^(3n+1) + 6n3^n+1 2^3n+1 · 3^9n+13^n+1 Step 5: Simplify by dividing terms with the same base using (a^m)/(a^n) = a^m-n. 2^3n+1 · 3^(9n+1) - (n+1) 2^3n+1 · 3^9n+1 - n - 1 2^3n+1 · 3^8n The simplified expression is 2^3n+1 · 3^8n. b) Simplify 5^2x+1 - 25^x5^2x+1 + 25^x Step 1: Express 25 as a power of 5. 25 = 5^2 Step 2: Substitute 25 = 5^2 into the expression. 5^2x+1 - (5^2)^x5^2x+1 + (5^2)^x 5^2x+1 - 5^2x5^2x+1 + 5^2x Step 3: Factor out the common term 5^2x from the numerator and denominator. Recall that 5^2x+1 = 5^2x · 5^1. 5^2x · 5 - 5^2x5^2x · 5 + 5^2x 5^2x(5 - 1)5^2x(5 + 1) Step 4: Cancel out the common factor 5^2x. (5 - 1)/(5 + 1) (4)/(6) Step 5: Simplify the fraction. (2)/(3) The simplified expression is (2)/(3). That's 2 down. 3 left today — send the next one.