Simplify the following expression: [(36)^(2n+1) (9)^n (4)^(2n)] / [(18)^n (2)^n (12)^(2n)]

Hey, good to see you again. Step 1: Express all bases as products of their prime factors. 36 = 2^2 × 3^2 9 = 3^2 4 = 2^2 18 = 2 × 3^2 12 = 2^2 × 3 Step 2: Substitute these prime factorizations into the expression. (2^2 × 3^2)^2n+1 × (3^2)^n × (2^2)^2n(2 × 3^2)^n × 2^n × (2^2 × 3)^2n Step 3: Apply the exponent rules (a^m)^n = a^mn and (ab)^m = a^m b^m. Numerator: (2^2 × 3^2)^2n+1 = 2^2(2n+1) × 3^2(2n+1) = 2^4n+2 × 3^4n+2 (3^2)^n = 3^2n (2^2)^2n = 2^4n Denominator: (2 × 3^2)^n = 2^n × 3^2n 2^n (2^2 × 3)^2n = 2^2(2n) × 3^2n = 2^4n × 3^2n Step 4: Rewrite the expression with the expanded terms. 2^4n+2 × 3^4n+2 × 3^2n × 2^4n2^n × 3^2n × 2^n × 2^4n × 3^2n Step 5: Combine terms with the same base in the numerator and denominator using the rule a^m × a^n = a^m+n. Numerator: 2^(4n+2) + 4n × 3^(4n+2) + 2n = 2^8n+2 × 3^6n+2 Denominator: 2^n+n+4n × 3^2n+2n = 2^6n × 3^4n Step 6: Divide the numerator by the denominator using the rule (a^m)/(a^n) = a^m-n. 2^8n+2 × 3^6n+22^6n × 3^4n = 2^(8n+2) - 6n × 3^(6n+2) - 4n = 2^2n+2 × 3^2n+2 Step 7: Combine the terms using the rule a^m × b^m = (ab)^m. (2 × 3)^2n+2 = 6^2n+2 The simplified expression is: 6^2n+2 Send me the next one 📸