11. Divide both sides by 2:
[tex]2\left(\dfrac19\right)^x=\dfrac2{81}\implies\left(\dfrac19\right)^x=\dfrac1{81}[/tex]
The solution has to be [tex]x=2[/tex] because
[tex]\left(\dfrac19\right)^2=\dfrac{1^2}{9^2}=\dfrac1{81}[/tex]
12. Divide both sides by 2:
[tex]2\left(\dfrac4{13}\right)^x=\dfrac{32}{169}\implies\left(\dfrac4{13}\right)^x=\dfrac{16}{169}[/tex]
On the right side we have two perfect squares:
[tex]\left(\dfrac4{13}\right)^x=\dfrac{4^2}{13^2}=\left(\dfrac4{13}\right)^2[/tex]
so again the answer is [tex]x=2[/tex].
14. Divide both sides by 8:
[tex]8\left(\dfrac23\right)^x=4\left(\dfrac{16}{27}\right)\implies\left(\dfrac23\right)^x=\dfrac8{27}[/tex]
On the right we have perfect cubes:
[tex]\left(\dfrac23\right)^x=\dfrac{2^3}{3^3}=\left(\dfrac23\right)^3[/tex]
so [tex]x=3[/tex].
15. [tex]\dfrac25\left(\dfrac25\right)^x=\dfrac8{125}[/tex]
We could divide both sides by 2/5 (or multiply both sides by 5/2, as the writing on your paper suggests). Then
[tex]\left(\dfrac25\right)^x=\dfrac{40}{250}=\dfrac4{25}[/tex]
The right side has two perfect squares:
[tex]\left(\dfrac25\right)^x=\dfrac{2^2}{5^2}=\left(\dfrac25\right)^2[/tex]
so that [tex]x=2[/tex].
Another way to do this is to rewrite the left side as
[tex]\dfrac25\left(\dfrac25\right)^x=\left(\dfrac25\right)^1\left(\dfrac25\right)^x=\left(\dfrac25\right)^{x+1}[/tex]
Meanwhile, on the right we have two perfect cubes:
[tex]\left(\dfrac25\right)^{x+1}=\dfrac{2^3}{5^3}=\left(\dfrac25\right)^3[/tex]
so that [tex]x+1=3[/tex], or [tex]x=2[/tex], as before.
