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

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Step 1: Express all bases as products of their prime factors. 36 = 2² × 3²; 9 = 3²; 4 = 2²; 18 = 2 × 3²; 12 = 2² × 3 Step 2: Substitute these prime factorizations into the expression. ((2² × 3²)²ⁿ⁺¹ × (3²)ⁿ × (2²)²ⁿ)/((2 × 3²)ⁿ × 2ⁿ × (2² × 3)²ⁿ) Step 3: Apply the exponent rules (a^m)ⁿ = a^(mn) and (ab)^m = a^m b^m. Numerator: (2² × 3²)²ⁿ⁺¹ = 2²⁽²ⁿ⁺¹⁾ × 3²⁽²ⁿ⁺¹⁾ = 2⁴ⁿ⁺² × 3⁴ⁿ⁺²; (3²)ⁿ = 3²ⁿ; (2²)²ⁿ = 2⁴ⁿ Denominator: (2 × 3²)ⁿ = 2ⁿ × 3²ⁿ 2ⁿ (2² × 3)²ⁿ = 2²⁽²ⁿ⁾ × 3²ⁿ = 2⁴ⁿ × 3²ⁿ Step 4: Rewrite the expression with the expanded terms. (2⁴ⁿ⁺² × 3⁴ⁿ⁺² × 3²ⁿ × 2⁴ⁿ)/(2ⁿ × 3²ⁿ × 2ⁿ × 2⁴ⁿ × 3²ⁿ) Step 5: Combine terms with the same base in the numerator and denominator using the rule a^m × aⁿ = a^(m+n). Numerator: 2⁽⁴ⁿ⁺²⁾⁺⁴ⁿ × 3⁽⁴ⁿ⁺²⁾⁺²ⁿ = 2⁸ⁿ⁺² × 3⁶ⁿ⁺² Denominator: 2ⁿ⁺ⁿ⁺⁴ⁿ × 3²ⁿ⁺²ⁿ = 2⁶ⁿ × 3⁴ⁿ; Step 6: Divide the numerator by the denominator using the rule (a^m)/(aⁿ) = a^(m−n). (2⁸ⁿ⁺² × 3⁶ⁿ⁺²)/(2⁶ⁿ × 3⁴ⁿ) = 2⁽⁸ⁿ⁺²⁾⁻⁶ⁿ × 3⁽⁶ⁿ⁺²⁾⁻⁴ⁿ; = 2²ⁿ⁺² × 3²ⁿ⁺²; Step 7: Combine the terms using the rule a^m × b^m = (ab)^m. (2 × 3)²ⁿ⁺² = 6²ⁿ⁺² The simplified expression is: 6²ⁿ⁺²

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

Answer

Step-by-step solution

Express all bases as products of their prime factors.

Substitute these prime factorizations into the expression.

Apply the exponent rules $(a^m)^n = a^{mn}$ and $(ab)^m = a^m b^m$ .

Rewrite the expression with the expanded terms.

Combine terms with the same base in the numerator and denominator using the rule $a^m \times a^n = a^{m+n}$ .

Divide the numerator by the denominator using the rule $\frac{a^m}{a^n} = a^{m-n}$ .

Combine the terms using the rule $a^m \times b^m = (ab)^m$ .

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

Answer

Step-by-step solution

Express all bases as products of their prime factors.

Substitute these prime factorizations into the expression.

Apply the exponent rules (am)n=amn(a^m)^n = a^{mn}(am)n=amn and (ab)m=ambm(ab)^m = a^m b^m(ab)m=ambm.

Rewrite the expression with the expanded terms.

Combine terms with the same base in the numerator and denominator using the rule am×an=am+na^m \times a^n = a^{m+n}am×an=am+n.

Divide the numerator by the denominator using the rule aman=am−n\frac{a^m}{a^n} = a^{m-n}anam​=am−n.

Combine the terms using the rule am×bm=(ab)ma^m \times b^m = (ab)^mam×bm=(ab)m.

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More Mathematics Questions

Apply the exponent rules $(a^m)^n = a^{mn}$ and $(ab)^m = a^m b^m$ .

Combine terms with the same base in the numerator and denominator using the rule $a^m \times a^n = a^{m+n}$ .

Divide the numerator by the denominator using the rule $\frac{a^m}{a^n} = a^{m-n}$ .

Combine the terms using the rule $a^m \times b^m = (ab)^m$ .