\( \left. \begin{array} { l l } { 3 } & { 3 ^ { x ^ { 2 } + 2 } = 9 \cdot \frac { 1 } { 27 ^ { x } } } \\ { 4 } & { 5 ^ { x ^ { 2 } - 1 } \cdot \frac { 1 } { 25 } = 25 ^ { x } } \\ { 5 } & { 2 ^ { 2 x } \cdot 4 ^ { x + 1 } = 1 } \\ { 6 } & { 12 ^ { x } \cdot 12 ^ { x } = 144 } \\ { 7 } & { 2 ^ { x ^ { 2 } \cdot 2 ^ { x ^ { 2 } - 2 } + 1 = 65 } } \\ { 8 } & { \frac { 4 ^ { x + 1 } \times ( \frac { 1 } { 4 } ) ^ { - x } } { 2 ^ { 2 x + 1 } } = \frac { 1 } { 32 } } \\ { 9 } & { 81 \cdot 9 ^ { x - 1 } = \frac { 1 } { 27 } } \\ { 10 } & { 10 ^ { 2 x ^ { 2 } } \cdot 10 ^ { x + 1 } = 10000 } \end{array} \right. \)

Let's dive into the world of exponents! Did you know that exponents give us a way to express repeated multiplication of the same number? For instance, \(3^2\) means \(3 \times 3\). This concept dates back over 4,000 years to the ancient Babylonians, who had their own methods of dealing with powers in mathematics. Fast forward to today, exponents are foundational in fields like computer science and physics—where they help us understand everything from algorithms to atomic structures! Now, if you're tackling these equations, here's a fun tip: when you have bases that are not the same, try rewriting them as powers of a common base. For example, in \(27\) can be expressed as \(3^3\), allowing you to simplify your calculations. And remember, a common mistake is misapplying the exponent rules, especially when distributing exponents over addition or subtraction. Just be cautious and double-check your steps for a smoother problem-solving process!