(b) \( 243^{x}=3^{2} \)

To solve \( 243^{x} = 3^{2} \), we start by expressing \( 243 \) as a power of \( 3 \). We know that \( 243 = 3^5 \), so we can rewrite the equation as \( (3^5)^{x} = 3^{2} \). This simplifies to \( 3^{5x} = 3^{2} \). Since the bases are the same, we can set the exponents equal to each other: \( 5x = 2 \). Dividing both sides by \( 5 \) gives us \( x = \frac{2}{5} \). Furthermore, it's useful to remember that exponents can simplify the process of solving equations. Understanding base conversions can help when working with different bases. For instance, if you encounter something like \( 8^y = 64 \), you might want to express both sides as powers of \( 2 \) (since \( 8 = 2^3 \) and \( 64 = 2^6 \)), making it easier to compare exponents. If you want to dive deeper into exponents, checking out resources on logarithms can be enlightening. Logarithms are the inverse of exponentiation and can simplify complex equations where standard exponent manipulation becomes cumbersome. Look for beginner-friendly books or online tutorials on logarithmic functions and their applications!