一、指数运算公式
1. 加法：
– \( a^m + a^n = (a^m) + (a^n) = a^{m+n} \)
– \( a^m \cdot a^n = a^{m+n} \)
2. 减法：
– \( a^m – a^n = (a^m) – (a^n) = a^{m-n} \)
– \( a^m \cdot a^n = a^{m+n} \)
3. 乘法：
– \( a^m \cdot a^n = a^{m+n} \)
– \( a^m \div a^n = a^{m-n} \)
4. 除法：
– \( a^m / a^n = a^{(m-n)/n} \)
– \( a^m \times a^n = a^{m+n} \)
5. 幂运算：
– \( a^m \cdot a^n = a^{m+n} \)
– \( a^m / a^n = a^{(m-n)/n} \)
– \( a^m \times a^n = a^{m+n} \)
6. 指数运算：
– \( a^m \cdot e^{k\ln a} = a^{m+k\ln a} \)
– \( a^m \cdot e^{-k\ln a} = a^{m-k\ln a} \)
– \( e^{k\ln a} / e^{-k\ln a} = e^{k(\ln a)^2} \)
7. 对数运算：
– \( e^{\ln a} = a \)
– \( e^{\ln b} = b \)
– \( \ln a / \ln b = \frac{\ln a}{\ln b} \)
– \( \ln a / (\ln b)^2 = \frac{1}{(\ln b)^2} \)
8. 三角函数：
– \(\sin(m\pi/2) = 1\)
– \(\cos(m\pi/2) = -1\)
– \(\tan(m\pi/2) = m\)
– \(\cot(m\pi/2) = 1/m\)
– \(\sec(m\pi/2) = 1/\sqrt{2}\)
– \(\csc(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\]
– \(\cscth(m\pi/2) = 1/m\)
– \(\sinh(m\pi/2) = 1/e\)
– \(\cosh(m\pi/2) = 1/e\)
– \(\tanh(m\pi/2) = m\)
– \(\coth(m\pi/2) = 1/m\)
– \(\sech(m\pi/2) = 1/\sqrt{e}\)
– \(\cseth(m\pi/2) = 1/\sqrt{e}\)$
二、解题指导：
1. 理解基本概念：首先确保你理解了指数运算的基本规则和性质，如加法、减法、乘法、除法等。
2. 记忆公式：熟练掌握各种指数运算的公式，如 $a^m + a^n = (a^m)^1 + (a^n)^1 = a^{m+n}$，$a^m \cdot a^n = a^{m+n}$，$a^m / a^n = a^{m-n}$ 等。
3. 应用公式：在解题时，根据题目要求选择合适的公式进行计算。例如，如果需要计算 $a^m$，可以直接使用 $a^m$；如果需要计算 $a^{m+n}$，可以使用 $(a^m)^1 + (a^n)^1$。
4. 注意负指数：当指数为负数时，需要注意其含义和计算方法。例如，$-a^n$ 表示 $a$ 的 $n$ 次方的倒数。
5. 化简表达式：在解题过程中，尽量将复杂的表达式化简为简单的形式，以便于计算和理解。
6. 检查答案：在完成计算后，要仔细检查答案，确保没有遗漏或错误。如果不确定，可以重新推导或查阅相关资料。