Answer
$x=\dfrac{22}{5}$
Work Step by Step
$\bf{\text{Solution Outline:}}$
To solve the given equation, $
16^{x+4}=8^{3x-2}
,$ use the laws of exponents to expressed both sides in the same base. Then equate the exponents. Use the properties of equality to isolate the variable.
$\bf{\text{Solution Details:}}$
Using exponents, the given equation is equivalent to
\begin{array}{l}\require{cancel}
(2^4)^{x+4}=(2^3)^{3x-2}
.\end{array}
Using the Power Rule of the laws of exponents which is given by $\left( x^m \right)^p=x^{mp},$ the equation above is equivalent to
\begin{array}{l}\require{cancel}
2^{4(x+4)}=2^{3(3x-2)}
.\end{array}
Since the bases are the same, then the exponents can be equated. Hence,
\begin{array}{l}\require{cancel}
4(x+4)=3(3x-2)
\\\\
4(x)+4(4)=3(3x)+3(-2)
\\\\
4x+16=9x-6
\\\\
4x-9x=-6-16
\\\\
-5x=-22
\\\\
x=\dfrac{-22}{-5}
\\\\
x=\dfrac{22}{5}
.\end{array}
