Question

1. Solve the following for x

8^2x-4=8^5x+1



2^x+6=16^3x+4



(1/2)^x=2^x+3



36^2x=216^3x-1







How much will a car be worth after 8 years if it depreciates in value by 12.6% each year?

Answer 1

Explanation:

`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=-(5)/(3)}\\\\================================`

`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=-(10)/(11)}\\\\================================`

`\left((1)/(2)\right)^x=2^(x+3)\qquad\text{use}\ a^(-1)=(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=-(3)/(2)}\\\\================================`

`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=(3)/(5)}\\\\================================`

`p\%=(p)/(100)\\\\100\%-12.6\%=87.4\%=(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.}`