RCISE 8 Solve for \( x \) : (a) \( \quad 4.3^{x}+3^{x}=15 \)

To solve the equation \( 4.3^{x} + 3^{x} = 15 \), we can start by letting \( y = 3^x \). Thus, we rewrite the equation as \( 4y + y = 15 \) which simplifies to \( 5y = 15 \) or \( y = 3 \). Now, substituting back, we have \( 3^x = 3 \). This implies that \( x = 1 \). So, the solution for \( x \) is \( x = 1 \). Now here's a bit of context: this kind of equation showcases the beauty of exponents, where they can turn into straightforward algebra by introducing substitutions! It’s a great demonstration of how we can simplify complex problems into manageable parts. If you like this problem, you might want to explore logarithms and their properties. They are fundamental tools in solving exponential equations and can lead to some interesting insights in calculus and beyond! Plus, they help tackle a variety of real-world problems ranging from population growth to compound interest calculations.