The equation given is:
[tex]2^{x+3}-2^x=k(2^x)[/tex]
We can use the property:
[tex]a^{b+c}=a^ba^c[/tex]
to break this apart. Shown below:
[tex]\begin{gathered} 2^{x+3}-2^x=k(2^x) \\ 2^x2^3-2^x=k(2^x) \end{gathered}[/tex]
We can solve for k [we divide both sides by 2^x to isolate k]:
[tex]\begin{gathered} 2^x2^3-2^x=k(2^x) \\ k=\frac{2^x2^3-2^x}{2^x} \end{gathered}[/tex]
Now, let's do a little algebra. The steps are shown below:
[tex]\begin{gathered} k=\frac{2^x2^3-2^x}{2^x} \\ k=\frac{2^x2^3}{2^x}-\frac{2^x}{2^x} \\ k=2^3-1 \\ k=8-1 \\ k=7 \end{gathered}[/tex]
The correct answer is
C
