here are the exponential rules:
[tex](ab)^c=(a^c)(b^c)[/tex]
[tex]\frac{1}{a^b}=a^{-b}[/tex]
[tex](a^b)^c=a^{bc}[/tex]
[tex](a^b)(a^c)=a^{b+c}[/tex]
if [tex]a^b=c^b[/tex] and b=b, then a=c
a.
she made a mistake in the 1st step because she used a positive exponent instead of a negative one for a fraction
in mathspeak: 1/16=1/(2^4)=2^-4, not 2^4
b.
[tex]2^{2x-3}=(\frac{1}{16})^{3x+9}[/tex]
[tex]2^{2x-3}=(\frac{1}{2^4})^{3x+9}[/tex]
[tex]2^{2x-3}=(2^{-4})^{3x+9}[/tex]
[tex]2^{2x-3}=2^{-4(3x+9)}[/tex]
[tex]2^{2x-3}=2^{-12x-36}[/tex]
[tex]2x-3=-12x-36[/tex]
[tex]-3=-14x-36[/tex]
[tex]33=-14x[/tex]
[tex]x=\frac{-33}{14}[/tex]
