2.2 .2 \quad 4^{x}-2^{x}=0

To solve the equation \( 4^{x}-2^{x}=0 \), we can rewrite \( 4^{x} \) as \( (2^{2})^{x} = 2^{2x} \). This simplifies our equation to \( 2^{2x} - 2^{x} = 0 \). Factoring out \( 2^{x} \), we get \( 2^{x}(2^{x} - 1) = 0 \). This gives us two cases: either \( 2^{x} = 0 \) (impossible, since \( 2^{x} > 0 \)) or \( 2^{x} - 1 = 0 \), leading to \( 2^{x} = 1 \). Therefore, solving \( 2^{x} = 1 \), we find \( x = 0 \). Did you know that exponential equations can lead to some fascinating real-world scenarios? For instance, they often model population growth—where you might find bacteria doubling every hour. By setting up equations similar to the one above, you can predict how long it will take for your bacterial colony to reach a certain size. Math isn’t just abstract; it can truly help us understand the living world! When working with exponential equations, a common mistake is forgetting to express bases in the same form. For example, mixing up 4 and 2, or neglecting to factor correctly, can lead to incorrect solutions. Always verify your steps and ensure you’re moving from the same base to accurately understand and solve the equation. Have fun with the numbers, and don’t rush through the process!