2. \( 5^{3 r}=5^{-2 r} \)

To solve the equation \( 5^{3r} = 5^{-2r} \), we can set the exponents equal to each other since the bases are the same: \[ 3r = -2r \] Now, adding \(2r\) from both sides gives: \[ 3r + 2r = 0 \] This simplifies to: \[ 5r = 0 \] Dividing both sides by 5 results in: \[ r = 0 \] Thus, the solution to the equation is \( r = 0 \). Now let’s dive into some interesting extra nuggets of knowledge! Did you know that exponents are actually shorthand for repeated multiplication? So, \( 5^{3r} \) means \( 5 \times 5 \times 5 \) multiplied by itself \( r \) times! It’s like a fun little math game where the rules become more complex but also more fascinating as you go along. Exponents have real-world applications in areas like finance and science! For instance, when calculating compound interest, the formula often involves exponents, showcasing how your investment grows exponentially over time. Similarly, in biology, the growth of populations can be modeled using exponential functions, illustrating just how far-reaching this concept is!