2) \( \frac{2.3^{x}+3^{x-2}}{5 \cdot 3^{x+1}-7 \cdot 2^{x-1}} \) 4) \( \frac{9^{x}+3^{x}-2}{9^{x}-4} \) 5) \( 4^{x}+3 \cdot 2^{x+1} \)
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Bonus Knowledge
When exploring functions like these, it's fascinating to note how exponential growth can dramatically differ based on the base of the exponent. For instance, the function \(4^x\) grows significantly faster than \(3^{x}\) for large values of \(x\) since 4 is greater than 3. This stark difference in growth rates can yield interesting behaviors in the equations when solved. In real-world applications, these types of exponential functions are foundational in fields such as finance, biology, and physics. For instance, \(9^x + 3^x - 2 = 0\) could model population growth, where \(9^x\) represents a population doubling every unit time. By analyzing these expressions, we can gain insights into growth patterns, optimize investments, or even predict trends in various scientific disciplines. So don't underestimate the power of exponents – they’re everywhere!
