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((1-tanh(2x)))/((1+tanh(2x)))=2

解答

\frac{( 1 - tanh ( 2 x ) )}{( 1 + tanh ( 2 x ) )} = 2

解答

x = - \frac{1}{4} ln (2)

+1

度数

x = - 9.92860 \dots^{\circ}

求解步骤

\frac{( 1 - tanh ( 2 x ) )}{( 1 + tanh ( 2 x ) )} = 2

使用三角恒等式改写

\frac{1 - tanh ( 2 x )}{1 + tanh ( 2 x )} = 2

使用双曲函数恒等式:

tanh (x) = \frac{e ^{x} - e ^{- x}}{e ^{x} + e ^{- x}}

\frac{1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}}{1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}} = 2

\frac{1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}}{1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}} = 2

\frac{1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}}{1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}} = 2 : x = - \frac{1}{4} ln (2)

\frac{1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}}{1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}} = 2

在两边乘以

1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}

\frac{1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}}{1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}} (1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}) = 2 (1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}})

化简

1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}} = 2 (1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}})

使用指数运算法则

1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}} = 2 (1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}})

使用指数法则:

a^{b c} = (a^{b})^{c}

e^{2 x} = (e^{x})^{2}, e^{- 2 x} = (e^{x})^{- 2}

1 - \frac{( e ^{x} ) ^{2} - ( e ^{x} ) ^{- 2}}{( e ^{x} ) ^{2} + ( e ^{x} ) ^{- 2}} = 2 (1 + \frac{( e ^{x} ) ^{2} - ( e ^{x} ) ^{- 2}}{( e ^{x} ) ^{2} + ( e ^{x} ) ^{- 2}})

1 - \frac{( e ^{x} ) ^{2} - ( e ^{x} ) ^{- 2}}{( e ^{x} ) ^{2} + ( e ^{x} ) ^{- 2}} = 2 (1 + \frac{( e ^{x} ) ^{2} - ( e ^{x} ) ^{- 2}}{( e ^{x} ) ^{2} + ( e ^{x} ) ^{- 2}})

用

e^{x} = u

改写方程式

1 - \frac{( u ) ^{2} - ( u ) ^{- 2}}{( u ) ^{2} + ( u ) ^{- 2}} = 2 (1 + \frac{( u ) ^{2} - ( u ) ^{- 2}}{( u ) ^{2} + ( u ) ^{- 2}})

解

1 - \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}} = 2 (1 + \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}}) : u = \frac{1}{4 2}, u = - \frac{1}{4 2}

1 - \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}} = 2 (1 + \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}})

整理后得

1 - \frac{u ^{4} - 1}{u ^{4} + 1} = 2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1})

在两边乘以

u^{4} + 1

1 - \frac{u ^{4} - 1}{u ^{4} + 1} = 2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1})

在两边乘以

u^{4} + 1

1 \cdot (u^{4} + 1) - \frac{u ^{4} - 1}{u ^{4} + 1} (u^{4} + 1) = 2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1)

化简

1 \cdot (u^{4} + 1) - \frac{u ^{4} - 1}{u ^{4} + 1} (u^{4} + 1) = 2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1)

化简

1 \cdot (u^{4} + 1) : u^{4} + 1

1 \cdot (u^{4} + 1)

乘以:

1 \cdot (u^{4} + 1) = (u^{4} + 1)

= (u^{4} + 1)

去除括号:

(a) = a

= u^{4} + 1

化简

- \frac{u ^{4} - 1}{u ^{4} + 1} (u^{4} + 1) : - (u^{4} - 1)

- \frac{u ^{4} - 1}{u ^{4} + 1} (u^{4} + 1)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{( u ^{4} - 1 ) ( u ^{4} + 1 )}{u ^{4} + 1}

约分:

u^{4} + 1

= - (u^{4} - 1)

u^{4} + 1 - (u^{4} - 1) = 2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1)

u^{4} + 1 - (u^{4} - 1) = 2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1)

u^{4} + 1 - (u^{4} - 1) = 2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1)

展开

u^{4} + 1 - (u^{4} - 1) : 2

u^{4} + 1 - (u^{4} - 1)

- (u^{4} - 1) : - u^{4} + 1

- (u^{4} - 1)

打开括号

= - (u^{4}) - (- 1)

使用加减运算法则

- (- a) = a, - (a) = - a

= - u^{4} + 1

= u^{4} + 1 - u^{4} + 1

化简

u^{4} + 1 - u^{4} + 1 : 2

u^{4} + 1 - u^{4} + 1

对同类项分组

= u^{4} - u^{4} + 1 + 1

同类项相加:

u^{4} - u^{4} = 0

= 1 + 1

数字相加:

1 + 1 = 2

= 2

= 2

展开

2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1) : 4 u^{4}

2 (1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1)

乘开

(1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1) : 2 u^{4}

(1 + \frac{u ^{4} - 1}{u ^{4} + 1}) (u^{4} + 1)

使用 FOIL 方法:

(a + b) (c + d) = a c + a d + b c + b d

a = 1, b = \frac{u ^{4} - 1}{u ^{4} + 1}, c = u^{4}, d = 1

= 1 \cdot u^{4} + 1 \cdot 1 + \frac{u ^{4} - 1}{u ^{4} + 1} u^{4} + \frac{u ^{4} - 1}{u ^{4} + 1} \cdot 1

= 1 \cdot u^{4} + 1 \cdot 1 + \frac{u ^{4} - 1}{u ^{4} + 1} u^{4} + 1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1}

化简

1 \cdot u^{4} + 1 \cdot 1 + \frac{u ^{4} - 1}{u ^{4} + 1} u^{4} + 1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1} : 2 u^{4}

1 \cdot u^{4} + 1 \cdot 1 + \frac{u ^{4} - 1}{u ^{4} + 1} u^{4} + 1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1}

1 \cdot u^{4} = u^{4}

1 \cdot u^{4}

乘以:

1 \cdot u^{4} = u^{4}

= u^{4}

1 \cdot 1 = 1

1 \cdot 1

数字相乘:

1 \cdot 1 = 1

= 1

\frac{u ^{4} - 1}{u ^{4} + 1} u^{4} = \frac{u ^{8} - u ^{4}}{u ^{4} + 1}

\frac{u ^{4} - 1}{u ^{4} + 1} u^{4}

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{( u ^{4} - 1 ) u ^{4}}{u ^{4} + 1}

乘开

(u^{4} - 1) u^{4} : u^{8} - u^{4}

(u^{4} - 1) u^{4}

= u^{4} (u^{4} - 1)

使用分配律:

a (b - c) = ab - a c

a = u^{4}, b = u^{4}, c = 1

= u^{4} u^{4} - u^{4} \cdot 1

= u^{4} u^{4} - 1 \cdot u^{4}

化简

u^{4} u^{4} - 1 \cdot u^{4} : u^{8} - u^{4}

u^{4} u^{4} - 1 \cdot u^{4}

u^{4} u^{4} = u^{8}

u^{4} u^{4}

使用指数法则:

a^{b} \cdot a^{c} = a^{b + c}

u^{4} u^{4} = u^{4 + 4}

= u^{4 + 4}

数字相加:

4 + 4 = 8

= u^{8}

1 \cdot u^{4} = u^{4}

1 \cdot u^{4}

乘以:

1 \cdot u^{4} = u^{4}

= u^{4}

= u^{8} - u^{4}

= u^{8} - u^{4}

= \frac{u ^{8} - u ^{4}}{u ^{4} + 1}

1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1} = \frac{u ^{4} - 1}{u ^{4} + 1}

1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1}

乘以:

1 \cdot \frac{u ^{4} - 1}{u ^{4} + 1} = \frac{u ^{4} - 1}{u ^{4} + 1}

= \frac{u ^{4} - 1}{u ^{4} + 1}

= u^{4} + 1 + \frac{u ^{8} - u ^{4}}{u ^{4} + 1} + \frac{u ^{4} - 1}{u ^{4} + 1}

合并分式

\frac{u ^{8} - u ^{4}}{u ^{4} + 1} + \frac{u ^{4} - 1}{u ^{4} + 1} : (u^{2} + 1) (u + 1) (u - 1)

使用法则

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{u ^{8} - u ^{4} + u ^{4} - 1}{u ^{4} + 1}

同类项相加:

- u^{4} + u^{4} = 0

= \frac{u ^{8} - 1}{u ^{4} + 1}

分解

u^{8} - 1 : (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) (u^{2} + 1) (u + 1) (u - 1)

u^{8} - 1

将

u^{8} - 1

改写为

(u^{4})^{2} - 1^{2}

u^{8} - 1

将

1

改写为

1^{2}

= u^{8} - 1^{2}

使用指数法则:

a^{b c} = (a^{b})^{c}

u^{8} = (u^{4})^{2}

= (u^{4})^{2} - 1^{2}

= (u^{4})^{2} - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(u^{4})^{2} - 1^{2} = (u^{4} + 1) (u^{4} - 1)

= (u^{4} + 1) (u^{4} - 1)

分解

u^{4} + 1 : (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

u^{4} + 1

u^{4} + 1 = (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

= (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

= (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) (u^{4} - 1)

分解

u^{4} - 1 : (u^{2} + 1) (u + 1) (u - 1)

u^{4} - 1

将

u^{4} - 1

改写为

(u^{2})^{2} - 1^{2}

u^{4} - 1

将

1

改写为

1^{2}

= u^{4} - 1^{2}

使用指数法则:

a^{b c} = (a^{b})^{c}

u^{4} = (u^{2})^{2}

= (u^{2})^{2} - 1^{2}

= (u^{2})^{2} - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

(u^{2})^{2} - 1^{2} = (u^{2} + 1) (u^{2} - 1)

= (u^{2} + 1) (u^{2} - 1)

分解

u^{2} - 1 : (u + 1) (u - 1)

u^{2} - 1

将

1

改写为

1^{2}

= u^{2} - 1^{2}

使用平方差公式:

x^{2} - y^{2} = (x + y) (x - y)

u^{2} - 1^{2} = (u + 1) (u - 1)

= (u + 1) (u - 1)

= (u^{2} + 1) (u + 1) (u - 1)

= (u^{2} + 2 u + 1) (u^{2} - 2 u + 1) (u^{2} + 1) (u + 1) (u - 1)

= \frac{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 ) ( u ^{2} + 1 ) ( u + 1 ) ( u - 1 )}{u ^{4} + 1}

u^{4} + 1 = (u^{2} + 2 u + 1) (u^{2} - 2 u + 1)

= \frac{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 ) ( u ^{2} + 1 ) ( u + 1 ) ( u - 1 )}{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}

消掉

\frac{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 ) ( u ^{2} + 1 ) ( u + 1 ) ( u - 1 )}{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )} : (u^{2} + 1) (u + 1) (u - 1)

\frac{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 ) ( u ^{2} + 1 ) ( u + 1 ) ( u - 1 )}{( u ^{2} + 2 u + 1 ) ( u ^{2} - 2 u + 1 )}

约分:

u^{2} + 2 u + 1

= \frac{( u ^{2} - 2 u + 1 ) ( u ^{2} + 1 ) ( u + 1 ) ( u - 1 )}{u ^{2} - 2 u + 1}

约分:

u^{2} - 2 u + 1

= (u^{2} + 1) (u + 1) (u - 1)

= (u^{2} + 1) (u + 1) (u - 1)

= u^{4} + 1 + (u^{2} + 1) (u + 1) (u - 1)

乘开

(u^{2} + 1) (u + 1) (u - 1) : u^{4} - 1

乘开

(u + 1) (u - 1) : u^{2} - 1

(u + 1) (u - 1)

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = u, b = 1

= u^{2} - 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

= u^{2} - 1

= (u^{2} + 1) (u^{2} - 1)

乘开

(u^{2} + 1) (u^{2} - 1) : u^{4} - 1

(u^{2} + 1) (u^{2} - 1)

使用平方差公式:

(a + b) (a - b) = a^{2} - b^{2}

a = u^{2}, b = 1

= (u^{2})^{2} - 1^{2}

化简

(u^{2})^{2} - 1^{2} : u^{4} - 1

(u^{2})^{2} - 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

= (u^{2})^{2} - 1

(u^{2})^{2} = u^{4}

(u^{2})^{2}

使用指数法则:

(a^{b})^{c} = a^{b c}

= u^{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= u^{4}

= u^{4} - 1

= u^{4} - 1

= u^{4} - 1

= u^{4} + 1 + u^{4} - 1

化简

u^{4} + 1 + u^{4} - 1 : 2 u^{4}

u^{4} + 1 + u^{4} - 1

对同类项分组

= u^{4} + u^{4} + 1 - 1

同类项相加:

u^{4} + u^{4} = 2 u^{4}

= 2 u^{4} + 1 - 1

1 - 1 = 0

= 2 u^{4}

= 2 u^{4}

= 2 u^{4}

= 2 \cdot 2 u^{4}

乘开

2 \cdot 2 u^{4} : 4 u^{4}

2 \cdot 2 u^{4}

打开括号

= 2 \cdot 2 u^{4}

数字相乘:

2 \cdot 2 = 4

= 4 u^{4}

= 4 u^{4}

2 = 4 u^{4}

解

2 = 4 u^{4} : u = \frac{1}{4 2}, u = - \frac{1}{4 2}

2 = 4 u^{4}

交换两边

4 u^{4} = 2

两边除以

4

4 u^{4} = 2

两边除以

4

\frac{4 u ^{4}}{4} = \frac{2}{4}

化简

u^{4} = \frac{1}{2}

u^{4} = \frac{1}{2}

对于

x^{n} = f (a)

，n 为偶数，解为

x = n f (a), - n f (a)

u = 4 \frac{1}{2}, u = - 4 \frac{1}{2}

4 \frac{1}{2} = \frac{1}{4 2}

4 \frac{1}{2}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b}, a \geq 0, b \geq 0

= \frac{4 1}{4 2}

使用根式运算法则:

n 1 = 1

41 = 1

= \frac{1}{4 2}

- 4 \frac{1}{2} = - \frac{1}{4 2}

- 4 \frac{1}{2}

使用根式运算法则:

n \frac{a}{b} = \frac{n a}{n b}, a \geq 0, b \geq 0

= - \frac{4 1}{4 2}

使用根式运算法则:

n 1 = 1

41 = 1

= - \frac{1}{4 2}

u = \frac{1}{4 2}, u = - \frac{1}{4 2}

u = \frac{1}{4 2}, u = - \frac{1}{4 2}

验证解

找到无定义的点（奇点）:

u = 0

取

1 - \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}}

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

取

2 (1 + \frac{u ^{2} - u ^{- 2}}{u ^{2} + u ^{- 2}})

的分母，令其等于零

解

u^{2} = 0 : u = 0

u^{2} = 0

使用法则

x^{n} = 0 \Rightarrow x = 0

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

u = \frac{1}{4 2}, u = - \frac{1}{4 2}

u = \frac{1}{4 2}, u = - \frac{1}{4 2}

代回

u = e^{x}

，求解

x

解

e^{x} = \frac{1}{4 2} : x = - \frac{1}{4} ln (2)

e^{x} = \frac{1}{4 2}

使用指数运算法则

e^{x} = \frac{1}{4 2}

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{4 2} = 2^{- \frac{1}{4}}

e^{x} = 2^{- \frac{1}{4}}

使用指数法则:

n a = a^{\frac{1}{n}}

2^{- \frac{1}{4}} = 2^{- \frac{1}{4}}

e^{x} = 2^{- \frac{1}{4}}

若

f (x) = g (x)

，则

ln (f (x)) = ln (g (x))

ln (e^{x}) = ln (2^{- \frac{1}{4}})

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

x = ln (2^{- \frac{1}{4}})

使用对数计算法则:

ln (x^{a}) = a \cdot ln (x)

ln (2^{- \frac{1}{4}}) = - \frac{1}{4} ln (2)

x = - \frac{1}{4} ln (2)

x = - \frac{1}{4} ln (2)

解

e^{x} = - \frac{1}{4 2} : x \in R

无解

e^{x} = - \frac{1}{4 2}

使用指数运算法则

e^{x} = - \frac{1}{4 2}

使用指数法则:

\frac{1}{a ^{b}} = a^{- b}

\frac{1}{4 2} = 2^{- \frac{1}{4}}

e^{x} = - 2^{- \frac{1}{4}}

e^{x} = - 2^{- \frac{1}{4}}

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

x = - \frac{1}{4} ln (2)

验证解:

x = - \frac{1}{4} ln (2)

真

将它们代入

\frac{1 - \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}}{1 + \frac{e ^{2 x} - e ^{- 2 x}}{e ^{2 x} + e ^{- 2 x}}} = 2

检验解是否符合

去除与方程不符的解。

代入

x = - \frac{1}{4} ln (2) :

真

\frac{1 - \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}{1 + \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}} = 2

\frac{1 - \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}{1 + \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}} = 2

\frac{1 - \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}{1 + \frac{e ^{2 (- \frac{1}{4} l n (2))} - e ^{- 2 (- \frac{1}{4} l n (2))}}{e ^{2 (- \frac{1}{4} l n (2))} + e ^{- 2 (- \frac{1}{4} l n (2))}}}

去除括号:

(- a) = - a, - (- a) = a

= \frac{1 - \frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}}{1 + \frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}} = - \frac{1}{3}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

乘

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{- \frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= (e^{l n (2)})^{- \frac{1}{2}}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2^{- \frac{1}{2}}

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{2}

e^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

乘

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{\frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= e^{l n (2)}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2

= \frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{\frac{1}{2} + 2}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

乘

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{- \frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= (e^{l n (2)})^{- \frac{1}{2}}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2^{- \frac{1}{2}}

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{2}

e^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

乘

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{\frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= e^{l n (2)}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2

= \frac{\frac{1}{2} - 2}{\frac{1}{2} + 2}

化简

\frac{1}{2} + 2 : \frac{3}{2}

\frac{1}{2} + 2

将项转换为分式:

2 = \frac{2 2}{2}

= \frac{1}{2} + \frac{2 2}{2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 2 2}{2}

1 + 22 = 3

1 + 22

使用根式运算法则:

a a = a

22 = 2

= 1 + 2

数字相加:

1 + 2 = 3

= 3

= \frac{3}{2}

= \frac{\frac{1}{2} - 2}{\frac{3}{2}}

化简

\frac{1}{2} - 2 : - \frac{1}{2}

\frac{1}{2} - 2

将项转换为分式:

2 = \frac{2 2}{2}

= \frac{1}{2} - \frac{2 2}{2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - 2 2}{2}

1 - 22 = - 1

1 - 22

使用根式运算法则:

a a = a

22 = 2

= 1 - 2

数字相减:

1 - 2 = - 1

= - 1

= \frac{- 1}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

= \frac{- \frac{1}{2}}{\frac{3}{2}}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{1}{2}}{\frac{3}{2}}

分式相除:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= - \frac{1 \cdot 2}{2 \cdot 3}

整理后得

= - \frac{2}{2 \cdot 3}

约分:

2

= - \frac{1}{3}

= \frac{1 - \frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}}{1 - \frac{1}{3}}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}} = - \frac{1}{3}

\frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{e ^{- 2 \cdot \frac{1}{4} l n (2)} + e ^{2 \cdot \frac{1}{4} l n (2)}}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

乘

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{- \frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= (e^{l n (2)})^{- \frac{1}{2}}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2^{- \frac{1}{2}}

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{2}

e^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

乘

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{\frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= e^{l n (2)}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2

= \frac{e ^{- 2 \cdot \frac{1}{4} l n (2)} - e ^{2 \cdot \frac{1}{4} l n (2)}}{\frac{1}{2} + 2}

e^{- 2 \cdot \frac{1}{4} l n (2)} = \frac{1}{2}

e^{- 2 \cdot \frac{1}{4} l n (2)}

乘

- 2 \cdot \frac{1}{4} ln (2) : - \frac{1}{2} ln (2)

- 2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= - \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{- \frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= (e^{l n (2)})^{- \frac{1}{2}}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2^{- \frac{1}{2}}

使用指数法则:

a^{- b} = \frac{1}{a ^{b}}

= \frac{1}{2}

e^{2 \cdot \frac{1}{4} l n (2)} = 2

e^{2 \cdot \frac{1}{4} l n (2)}

乘

2 \cdot \frac{1}{4} ln (2) : \frac{1}{2} ln (2)

2 \cdot \frac{1}{4} ln (2)

分式相乘:

a \cdot \frac{b}{c} = \frac{a \cdot b}{c}

= \frac{1 \cdot 2}{4} ln (2)

\frac{1 \cdot 2}{4} = \frac{1}{2}

\frac{1 \cdot 2}{4}

数字相乘:

1 \cdot 2 = 2

= \frac{2}{4}

约分:

2

= \frac{1}{2}

= \frac{1}{2}

= e^{\frac{1}{2} l n (2)}

使用指数法则:

a^{b c} = (a^{b})^{c}

= e^{l n (2)}

使用对数计算法则:

a^{l o g_{a} (b)} = b

e^{l n (2)} = 2

= 2

= \frac{\frac{1}{2} - 2}{\frac{1}{2} + 2}

化简

\frac{1}{2} + 2 : \frac{3}{2}

\frac{1}{2} + 2

将项转换为分式:

2 = \frac{2 2}{2}

= \frac{1}{2} + \frac{2 2}{2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 + 2 2}{2}

1 + 22 = 3

1 + 22

使用根式运算法则:

a a = a

22 = 2

= 1 + 2

数字相加:

1 + 2 = 3

= 3

= \frac{3}{2}

= \frac{\frac{1}{2} - 2}{\frac{3}{2}}

化简

\frac{1}{2} - 2 : - \frac{1}{2}

\frac{1}{2} - 2

将项转换为分式:

2 = \frac{2 2}{2}

= \frac{1}{2} - \frac{2 2}{2}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 - 2 2}{2}

1 - 22 = - 1

1 - 22

使用根式运算法则:

a a = a

22 = 2

= 1 - 2

数字相减:

1 - 2 = - 1

= - 1

= \frac{- 1}{2}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{1}{2}

= \frac{- \frac{1}{2}}{\frac{3}{2}}

使用分式法则:

\frac{- a}{b} = - \frac{a}{b}

= - \frac{\frac{1}{2}}{\frac{3}{2}}

分式相除:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= - \frac{1 \cdot 2}{2 \cdot 3}

整理后得

= - \frac{2}{2 \cdot 3}

约分:

2

= - \frac{1}{3}

= \frac{1 - ( - \frac{1}{3} )}{1 - \frac{1}{3}}

化简

\frac{1 - ( - \frac{1}{3} )}{1 - \frac{1}{3}}

使用法则

- (- a) = a

= \frac{1 + \frac{1}{3}}{1 - \frac{1}{3}}

化简

1 - \frac{1}{3} : \frac{2}{3}

1 - \frac{1}{3}

将项转换为分式:

1 = \frac{1 \cdot 3}{3}

= \frac{1 \cdot 3}{3} - \frac{1}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 3 - 1}{3}

1 \cdot 3 - 1 = 2

1 \cdot 3 - 1

数字相乘:

1 \cdot 3 = 3

= 3 - 1

数字相减:

3 - 1 = 2

= 2

= \frac{2}{3}

= \frac{1 + \frac{1}{3}}{\frac{2}{3}}

化简

1 + \frac{1}{3} : \frac{4}{3}

1 + \frac{1}{3}

将项转换为分式:

1 = \frac{1 \cdot 3}{3}

= \frac{1 \cdot 3}{3} + \frac{1}{3}

因为分母相等，所以合并分式:

\frac{a}{c} \pm \frac{b}{c} = \frac{a \pm b}{c}

= \frac{1 \cdot 3 + 1}{3}

1 \cdot 3 + 1 = 4

1 \cdot 3 + 1

数字相乘:

1 \cdot 3 = 3

= 3 + 1

数字相加:

3 + 1 = 4

= 4

= \frac{4}{3}

= \frac{\frac{4}{3}}{\frac{2}{3}}

分式相除:

\frac{\frac{a}{b}}{\frac{c}{d}} = \frac{a \cdot d}{b \cdot c}

= \frac{4 \cdot 3}{3 \cdot 2}

约分:

3

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

= 2

2 = 2

真

解是

x = - \frac{1}{4} ln (2)

x = - \frac{1}{4} ln (2)

作图

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