Solve for \( x: 8^{3 x+2}=16^{3 x-1} \) \[ \begin{array}{l}x=\frac{4}{3} \\ O-\frac{10}{3} \\ \\ \text { Question Help: Video }\end{array} \]

To solve for \( x \) in the equation \( 8^{3x+2} = 16^{3x-1} \), we can start by rewriting the bases as powers of 2. We know that \( 8 = 2^3 \) and \( 16 = 2^4 \). This allows us to rewrite the equation as: \[ (2^3)^{3x+2} = (2^4)^{3x-1} \] This simplifies to: \[ 2^{9x+6} = 2^{12x-4} \] Now that the bases are the same, we can set the exponents equal to each other: \[ 9x + 6 = 12x - 4 \] Rearranging gives us: \[ 6 + 4 = 12x - 9x \] \[ 10 = 3x \] Dividing by 3 yields: \[ x = \frac{10}{3} \] Thus, the solution to the equation \( 8^{3x+2}=16^{3x-1} \) is \[ x = \frac{10}{3}. \] In terms of fun facts about the numbers involved, did you know that exponents can transform and simplify many problems? It's like finding secret shortcuts in math! Another interesting angle is the crucial role of exponent rules in many real-world applications, like calculating interest rates or understanding exponential growth in populations. They help us make sense of how things evolve over time!