Step-by-step explanation:
[tex]8^{2x-4}=8^{5x+1}\iff2x-4=5x+1\qquad\text{add 4 to both sides}\\\\2x=5x+5\qquad\text{subtract 5x from both sides}\\\\-3x=5\qquad\text{divide both sides by (-3)}\\\\\boxed{x=-\dfrac{5}{3}}\\\\================================[/tex]
[tex]2^{x+6}=16^{3x+4}\\\\2^{x+6}=(2^4)^{3x+4}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\2^{x+6}=2^{4(3x+4)}\iff x+6=4(3x+4)\qquad\text{use the distributive property}\\\\x+6=(4)(3x)+(4)(4)\\\\x+6=12x+16\qquad\text{subtract 6 from both sides}\\\\x=12x+10\qquad\text{subtract 12x from both sides}\\\\-11x=10\qquad\text{divide both sides by (-11)}\\\\\boxed{x=-\dfrac{10}{11}}\\\\================================[/tex]
[tex]\left(\dfrac{1}{2}\right)^x=2^{x+3}\qquad\text{use}\ a^{-1}=\dfrac{1}{a}\\\\(2^{-1})^x=2^{x+3}\\\\2^{-x}=2^{x+3}\iff -x=x+3\qquad\text{subtract x from both sides}\\\\-2x=3\qquad\text{divide both sides by (-2)}\\\\\boxed{x=-\dfrac{3}{2}}\\\\================================[/tex]
[tex]36^{2x}=216^{3x-1}\\\\(6^2)^{2x}=(6^3)^{3x-1}\qquad\text{use}\ (a^n)^m=a^{nm}\\\\6^{(2)(2x)}=2^{3(3x-1)}\iff(2)(2x)=3(3x-1)\qquad\text{use the distributive property}\\\\4x=(3)(3x)+(3)(-1)\\\\4x=9x-3\qquad\text{subtract 9x from both sides}\\\\-5x=-3\qquad\text{divide both sides by (-5)}\\\\\boxed{x=\dfrac{3}{5}}\\\\================================[/tex]
[tex]p\%=\dfrac{p}{100}\\\\100\%-12.6\%=87.4\%=\dfrac{87.4}{100}=0.874\\\\8\ years\to(0.874)^4\approx0.584\to58.4\%\\\\\text{After 8 years, the car will be worth 58.4}\%\ \text{of the initial price.}[/tex]
