Question

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Answer 1

c, \(a^{\left(n+5\right)\left(n-8\right)}=1=a^0\Rightarrow\left(n+5\right)\left(n-8\right)=0\Leftrightarrow\left[{}\begin{matrix}n=-5\\n=8\end{matrix}\right.\)

d, \(\dfrac{1}{2}.2^n=2.3^2.4^2-4.2^n\)

\(\Leftrightarrow2^{n-1}=288-4.2^n=4\left(72-2^n\right)\)

\(\Leftrightarrow2^{n-3}=-2^n-72\Leftrightarrow2^{n-3}+2^n=72\Rightarrow n=6\)

 

Answer 2

\(\text{c) }a^{\left(n+5\right)\left(n-8\right)}=1\\ \Rightarrow\left(n+5\right)\left(n-8\right)=0\\ \Rightarrow\left[{}\begin{matrix}n+5=0\\n-8=0\end{matrix}\right.\Rightarrow\left[{}\begin{matrix}n=-5\\n=8\end{matrix}\right.\)

\(\text{d) }\dfrac{1}{2}\cdot2^n=2^1\cdot3^2\cdot4^2-4\cdot2^n\\ \Rightarrow\dfrac{2^n}{2}=3^2\cdot2\cdot\left(2^2\right)^2-2^2\cdot2^n\\ \Rightarrow2^n=3^2\cdot2^2\cdot2^4-2^3\cdot2^n\\ \Rightarrow2^n+2^n\cdot8=9\cdot2^6\\ \Rightarrow2^n\cdot\left(1+8\right)=9\cdot2^6\\ \Rightarrow2^n\cdot9=2^6\cdot9\\ \Rightarrow n=6\)

#$\mathtt{Toru}$