3.1.1 \( \frac{7^{x+1}-7^{x}}{6.7^{x}} \)

Let's simplify your expression step by step! First, notice that you can rewrite \( 7^{x+1} \) as \( 7^x \cdot 7 \). This takes your expression to: \[ \frac{7 \cdot 7^x - 7^x}{6 \cdot 7^x} \] Now, factoring out \( 7^x \) from the numerator gives us: \[ \frac{7^x (7 - 1)}{6 \cdot 7^x} = \frac{6 \cdot 7^x}{6 \cdot 7^x} = 1 \] So, your expression simplifies nicely to just \( 1 \)! Isn't it fascinating how algebra can turn what looks like a complicated expression into a simple number? Simplifying expressions helps unlock the beauty of math, revealing how everything is interconnected, just like a puzzle where each piece fits perfectly together! Plus, it’s a great skill for solving real-world problems, like finding quantities in finance or computing areas in geometry.