Answer: [tex]x=\frac{37}{9}[/tex]
Step-by-step explanation:
By the negative exponent rule, you have that:
[tex](\frac{1}{a})^n=a^{-n}[/tex]
By the exponents properties, you know that:
[tex](m^n)^l=m^{(nl)}[/tex]
[tex](m^n)(m^l)=m^{(n+l)}[/tex]
Rewrite 4, 8 and 32 as following:
4=2²
8=2³
32=2⁵
Rewrite the expression:
[tex](2^2)^{(x-7)}*(2^3)^{(2x-3)}=\frac{32}{2^{(x-9)}}[/tex]
Keeping on mind the exponents properties, you have:
[tex](2)^{2(x-7)}*(2)^{3(2x-3)}=32(2^{-(x-9)}[/tex]
[tex](2)^{2(x-7)}*(2)^{3(2x-3)}=(2^5)(2^{-(x-9)})\\\\(2)^{(2x-14)}*(2)^{(6x-9)}=(2^5)(2^{(-x+9)})\\\\2^{((2x-14)+(6x-9))}=2^{(5+(-x+9))}[/tex]
As the bases are equal, then:
[tex](2x-14)+(6x-9)=5+(-x+9)\\\\2x-14+6x-9=5-x+9\\\\8x-23=14-x\\9x=37[/tex]
[tex]x=\frac{37}{9}[/tex]
