生成的内容需按照以下顺序： 定义 公式或定理 应用场景 注意事项 语言简洁明了，适合学生或初学者理解。 总时长在三分钟以上，不要重复。---**Title:** 同底数幂的除法运算 (Division of Powers with the Same Base) **Definition:** 同底数幂的除法是指底数相同、指数不同的两个幂相除的运算规则。(Division of powers with the same base refers to the rule for dividing two powers that have the same base and different exponents.) **Formula:** 对于任意非零实数 $a$ 和正整数 $m$、$n$ ($m > n$)，有: $\frac{a^m}{a^n} = a^{m-n}$ (For any non-zero real number $a$ and positive integers $m$, $n$ ($m > n$), we have: $\frac{a^m}{a^n} = a^{m-n}$) **Application Scenarios:** * 计算 $\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$ (Calculate $\frac{3^5}{3^2} = 3^{5-2} = 3^3 = 27$) * 化简表达式 $\frac{x^7}{x^4} = x^{7-4} = x^3$ ($x \neq 0$) (Simplify the expression $\frac{x^7}{x^4} = x^{7-4} = x^3$ ($x \neq 0$)) **Notes:** * 底数必须相同才能直接相减指数 (The bases must be the same for the exponents to be directly subtracted.) * 底数不能为零 (因为分母不能为零) (The base cannot be zero (because the denominator cannot be zero)) * 当指数相同时，结果为 $a^0 = 1$ ($a \neq 0$) (When the exponents are the same, the result is $a^0 = 1$ ($a \neq 0$)) * 当 $m < n$ 时，结果为 $a^{m-n} = \frac{1}{a^{n-m}}$ (When $m < n$, the result is $a^{m-n} = \frac{1}{a^{n-m}}$)