Answer:
[tex]\dfrac{3^{13}}{2}[/tex]
Step-by-step explanation:
Given expression:
[tex]\dfrac{3^5 \cdot 2^4 \cdot 2^1 \cdot 3^6 \cdot (3^2)^4}{(2^3)^2 \cdot 3^6}[/tex]
[tex]\textsf{Apply exponent rule} \quad (a^b)^c=a^{bc}:[/tex]
[tex]\implies \dfrac{3^5 \cdot 2^4 \cdot 2^1 \cdot 3^6 \cdot 3^8}{2^6 \cdot 3^6}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^b \cdot a^c=a^{b+c}:[/tex]
[tex]\implies \dfrac{3^{(5+6+8)} \cdot 2^{(4+1)}}{2^6 \cdot 3^6}[/tex]
[tex]\implies \dfrac{3^{19} \cdot 2^{5}}{2^6 \cdot 3^6}[/tex]
[tex]\implies \dfrac{3^{19} \cdot 2^{5}}{3^6 \cdot 2^6}[/tex]
[tex]\textsf{Apply exponent rule} \quad \dfrac{a^b}{a^c}=a^{b-c}:[/tex]
[tex]\implies 3^{(19-6)} \cdot 2^{(5-6)}[/tex]
[tex]\implies 3^{13} \cdot 2^{-1}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^{-n}=\dfrac{1}{a^n}:[/tex]
[tex]\implies \dfrac{3^{13}}{2^1}[/tex]
[tex]\textsf{Apply exponent rule} \quad a^1=a:[/tex]
[tex]\implies \dfrac{3^{13}}{2}[/tex]
