Group terms with the same base.

Here are the step-by-step solutions for each expression: i) Step 1: Group terms with the same base. 5^2n+15^2n × 3^2n-33^2n Step 2: Apply the quotient rule for exponents, (a^m)/(a^n) = a^m-n. 5^(2n+1) - 2n × 3^(2n-3) - 2n Step 3: Simplify the exponents. 5^1 × 3^-3 Step 4: Evaluate the powers. 5 × (1)/(3^3) = 5 × (1)/(27) = (5)/(27) The simplified expression is (5)/(27). j) Step 1: Combine terms in the numerator with base 5 using a^m × a^n = a^m+n. Note that 5 = 5^1. 5^2n+1 + (2n-2) + 125^2n-1 = 5^4n25^2n-1 Step 2: Express the denominator with base 5, since 25 = 5^2. 5^4n(5^2)^2n-1 Step 3: Apply the power of a power rule, (a^m)^n = a^mn. 5^4n5^2(2n-1) = 5^4n5^4n-2 Step 4: Apply the quotient rule for exponents, (a^m)/(a^n) = a^m-n. 5^4n - (4n-2) = 5^4n - 4n + 2 = 5^2 Step 5: Evaluate the power. 25 The simplified expression is 25. k) Step 1: Express all bases as products of their prime factors: 12 = 2^2 × 3, 9 = 3^2, 36 = 2^2 × 3^2, 8 = 2^3. (2^2 × 3)^n+1 × (3^2)^2n-1(2^2 × 3^2)^n × (2^3)^1-n Step 2: Apply the power rules (ab)^m = a^m b^m and (a^m)^n = a^mn. 2^2(n+1) × 3^n+1 × 3^2(2n-1)2^2n × 3^2n \