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64cosh^4(x)-64cosh^2(x)-9=0

解答

64 cosh^{4} (x) - 64 cosh^{2} (x) - 9 = 0

解答

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

+1

度数

x = 19.85720 \dots^{\circ}, x = - 19.85720 \dots^{\circ}

求解步骤

64 cosh^{4} (x) - 64 cosh^{2} (x) - 9 = 0

使用三角恒等式改写

64 cosh^{4} (x) - 64 cosh^{2} (x) - 9 = 0

使用双曲函数恒等式:

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

64 (\frac{e ^{x} + e ^{- x}}{2})^{4} - 64 (\frac{e ^{x} + e ^{- x}}{2})^{2} - 9 = 0

64 (\frac{e ^{x} + e ^{- x}}{2})^{4} - 64 (\frac{e ^{x} + e ^{- x}}{2})^{2} - 9 = 0

64 (\frac{e ^{x} + e ^{- x}}{2})^{4} - 64 (\frac{e ^{x} + e ^{- x}}{2})^{2} - 9 = 0 : x = \frac{1}{2} ln (2), x = - \frac{1}{2} ln (2)

64 (\frac{e ^{x} + e ^{- x}}{2})^{4} - 64 (\frac{e ^{x} + e ^{- x}}{2})^{2} - 9 = 0

使用指数运算法则

64 (\frac{e ^{x} + e ^{- x}}{2})^{4} - 64 (\frac{e ^{x} + e ^{- x}}{2})^{2} - 9 = 0

使用指数法则:

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

e^{- x} = (e^{x})^{- 1}

64 (\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{4} - 64 (\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{2} - 9 = 0

64 (\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{4} - 64 (\frac{e ^{x} + ( e ^{x} ) ^{- 1}}{2})^{2} - 9 = 0

用

e^{x} = u

改写方程式

64 (\frac{u + ( u ) ^{- 1}}{2})^{4} - 64 (\frac{u + ( u ) ^{- 1}}{2})^{2} - 9 = 0

解

64 (\frac{u + u ^{- 1}}{2})^{4} - 64 (\frac{u + u ^{- 1}}{2})^{2} - 9 = 0 : u = - \frac{1}{2}, u = - 2, u = 2, u = \frac{1}{2}

64 (\frac{u + u ^{- 1}}{2})^{4} - 64 (\frac{u + u ^{- 1}}{2})^{2} - 9 = 0

分解

64 (\frac{u + u ^{- 1}}{2})^{4} - 64 (\frac{u + u ^{- 1}}{2})^{2} - 9 : (\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1) (2 (u + \frac{1}{u}) + 3) (2 (u + \frac{1}{u}) - 3)

64 (\frac{u + u ^{- 1}}{2})^{4} - 64 (\frac{u + u ^{- 1}}{2})^{2} - 9

令

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

= 64 u^{2} - 64 u - 9

分解

64 u^{2} - 64 u - 9 : (8 u + 1) (8 u - 9)

64 u^{2} - 64 u - 9

将表达式拆分成组

64 u^{2} - 64 u - 9

定义

576

的因数:

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576

576

约数 (因数)

找到

576

的质因数:

2, 2, 2, 2, 2, 2, 3, 3

576

576

除以

2576 = 288 \cdot 2

= 2 \cdot 288

288

除以

2288 = 144 \cdot 2

= 2 \cdot 2 \cdot 144

144

除以

2144 = 72 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 72

72

除以

272 = 36 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 36

36

除以

236 = 18 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 18

18

除以

218 = 9 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 9

9

除以

39 = 3 \cdot 3

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

乘以

576

的质因数:

4, 8, 16, 32, 64, 6, 12, 24, 48, 96, 192, 9, 18, 36, 72, 144, 288

2 \cdot 2 = 4

2 \cdot 2 \cdot 2 = 8

4, 8, 16, 32, 64, 6, 12, 24, 48, 96, 192, 9, 18, 36, 72, 144, 288

4, 8, 16, 32, 64, 6, 12, 24, 48, 96, 192, 9, 18, 36, 72, 144, 288

添加质因数:

2, 3

将 1 和数字

576

自身相加

1, 576

576

的因数

1, 2, 3, 4, 6, 8, 9, 12, 16, 18, 24, 32, 36, 48, 64, 72, 96, 144, 192, 288, 576

576

的负因数:

- 1, - 2, - 3, - 4, - 6, - 8, - 9, - 12, - 16, - 18, - 24, - 32, - 36, - 48, - 64, - 72, - 96, - 144, - 192, - 288, - 576

将因数乘以

- 1

得到负因数

- 1, - 2, - 3, - 4, - 6, - 8, - 9, - 12, - 16, - 18, - 24, - 32, - 36, - 48, - 64, - 72, - 96, - 144, - 192, - 288, - 576

对于每两个因数

u * v = - 576

，检验是否

u + v = - 64

检验

u = 1, v = - 576 : u * v = - 576, u + v = - 575 \Rightarrow

假检验

u = 2, v = - 288 : u * v = - 576, u + v = - 286 \Rightarrow

假

u = 8, v = - 72

分组为

(a x^{2} + ux) + (vx + c)

(64 u^{2} + 8 u) + (- 72 u - 9)

= (64 u^{2} + 8 u) + (- 72 u - 9)

从

64 u^{2} + 8 u

分解出因式

8 u

:

8 u (8 u + 1)

64 u^{2} + 8 u

使用指数法则:

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

u^{2} = uu

= 64 uu + 8 u

将

64

改写为

8 \cdot 8

= 8 \cdot 8 uu + 8 u

因式分解出通项

8 u

= 8 u (8 u + 1)

从

- 72 u - 9

分解出因式

- 9

:

- 9 (8 u + 1)

- 72 u - 9

将

72

改写为

9 \cdot 8

= - 9 \cdot 8 u - 9

因式分解出通项

- 9

= - 9 (8 u + 1)

= 8 u (8 u + 1) - 9 (8 u + 1)

因式分解出通项

8 u + 1

= (8 u + 1) (8 u - 9)

= (8 u + 1) (8 u - 9)

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

代回

= (8 (\frac{u + u ^{- 1}}{2})^{2} + 1) (8 (\frac{u + u ^{- 1}}{2})^{2} - 9)

分解

8 (\frac{u + u ^{- 1}}{2})^{2} - 9 : (8 \frac{u + u ^{- 1}}{2} + 3) (8 \frac{u + u ^{- 1}}{2} - 3)

8 (\frac{u + u ^{- 1}}{2})^{2} - 9

将

8 (\frac{u + u ^{- 1}}{2})^{2} - 9

改写为

(8 \frac{u + u ^{- 1}}{2})^{2} - 3^{2}

8 (\frac{u + u ^{- 1}}{2})^{2} - 9

使用根式运算法则:

a = (a)^{2}

8 = (8)^{2}

= (8)^{2} (\frac{u + u ^{- 1}}{2})^{2} - 9

将

9

改写为

3^{2}

= (8)^{2} (\frac{u + u ^{- 1}}{2})^{2} - 3^{2}

使用指数法则:

a^{m} b^{m} = (ab)^{m}

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

= (8 (\frac{u + u ^{- 1}}{2}))^{2} - 3^{2}

= (8 (\frac{u + u ^{- 1}}{2}))^{2} - 3^{2}

使用平方差公式:

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

(8 \frac{u + u ^{- 1}}{2})^{2} - 3^{2} = (8 \frac{u + u ^{- 1}}{2} + 3) (8 \frac{u + u ^{- 1}}{2} - 3)

= (8 (\frac{u + u ^{- 1}}{2}) + 3) (8 (\frac{u + u ^{- 1}}{2}) - 3)

= (8 (\frac{u + u ^{- 1}}{2})^{2} + 1) (8 (\frac{u + u ^{- 1}}{2}) + 3) (8 (\frac{u + u ^{- 1}}{2}) - 3)

整理后得

= (8 (\frac{u + u ^{- 1}}{2})^{2} + 1) (22 (\frac{u + u ^{- 1}}{2}) + 3) (22 (\frac{u + u ^{- 1}}{2}) - 3)

化简

(8 (\frac{u + u ^{- 1}}{2})^{2} + 1) (22 (\frac{u + u ^{- 1}}{2}) + 3) (22 (\frac{u + u ^{- 1}}{2}) - 3) : (\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1) (2 (u + \frac{1}{u}) + 3) (2 (u + \frac{1}{u}) - 3)

(8 (\frac{u + u ^{- 1}}{2})^{2} + 1) (22 (\frac{u + u ^{- 1}}{2}) + 3) (22 (\frac{u + u ^{- 1}}{2}) - 3)

去除括号:

(a) = a

= (8 (\frac{u + u ^{- 1}}{2})^{2} + 1) (22 \frac{u + u ^{- 1}}{2} + 3) (22 \frac{u + u ^{- 1}}{2} - 3)

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

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

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

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

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

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

使用指数法则:

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

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

化简

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

u + \frac{1}{u}

将项转换为分式:

u = \frac{uu}{u}

= \frac{uu}{u} + \frac{1}{u}

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{uu + 1}{u}

uu + 1 = u^{2} + 1

uu + 1

uu = u^{2}

uu

使用指数法则:

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

uu = u^{1 + 1}

= u^{1 + 1}

数字相加:

1 + 1 = 2

= u^{2}

= u^{2} + 1

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

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

使用分式法则:

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

= \frac{u ^{2} + 1}{u \cdot 2}

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

使用指数法则:

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

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

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

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

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

= 8 \cdot \frac{( u ^{2} + 1 ) ^{2}}{2 ^{2} u ^{2}}

分式相乘:

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

= \frac{( u ^{2} + 1 ) ^{2} \cdot 8}{u ^{2} \cdot 2 ^{2}}

分解

8 : 2^{3}

因式分解

8 = 2^{3}

= \frac{2 ^{3} ( u ^{2} + 1 ) ^{2}}{2 ^{2} u ^{2}}

消掉

\frac{( u ^{2} + 1 ) ^{2} \cdot 2 ^{3}}{u ^{2} \cdot 2 ^{2}} : \frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}}

\frac{( u ^{2} + 1 ) ^{2} \cdot 2 ^{3}}{u ^{2} \cdot 2 ^{2}}

使用指数法则:

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

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

= \frac{2 ^{3 - 2} ( u ^{2} + 1 ) ^{2}}{u ^{2}}

数字相减:

3 - 2 = 1

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

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

= (\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1) (22 \frac{u ^{- 1} + u}{2} + 3) (22 \frac{u ^{- 1} + u}{2} - 3)

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

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

分式相乘:

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

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

约分:

2

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

使用指数法则:

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

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

= (\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1) (2 (u + \frac{1}{u}) + 3) (22 \frac{u ^{- 1} + u}{2} - 3)

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

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

分式相乘:

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

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

约分:

2

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

使用指数法则:

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

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

= (\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1) (2 (u + \frac{1}{u}) + 3) (2 (u + \frac{1}{u}) - 3)

= (\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1) (2 (u + \frac{1}{u}) + 3) (2 (u + \frac{1}{u}) - 3)

(\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1) (2 (u + \frac{1}{u}) + 3) (2 (u + \frac{1}{u}) - 3) = 0

使用零因数法则： If

ab = 0

then

a = 0

or

b = 0

\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1 = 0 or 2 (u + \frac{1}{u}) + 3 = 0 or 2 (u + \frac{1}{u}) - 3 = 0

解

\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1 = 0 : u \in R

无解

\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1 = 0

在两边乘以

u^{2}

\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} + 1 = 0

在两边乘以

u^{2}

\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} u^{2} + 1 \cdot u^{2} = 0 \cdot u^{2}

化简

\frac{2 ( u ^{2} + 1 ) ^{2}}{u ^{2}} u^{2} + 1 \cdot u^{2} = 0 \cdot u^{2}

化简

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

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

分式相乘:

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

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

约分:

u^{2}

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

化简

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

1 \cdot u^{2}

乘以:

1 \cdot u^{2} = u^{2}

= u^{2}

化简

0 \cdot u^{2} : 0

0 \cdot u^{2}

使用法则

0 \cdot a = 0

= 0

2 (u^{2} + 1)^{2} + u^{2} = 0

2 (u^{2} + 1)^{2} + u^{2} = 0

2 (u^{2} + 1)^{2} + u^{2} = 0

解

2 (u^{2} + 1)^{2} + u^{2} = 0 : u \in R

无解

2 (u^{2} + 1)^{2} + u^{2} = 0

展开

2 (u^{2} + 1)^{2} + u^{2} : 2 u^{4} + 5 u^{2} + 2

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

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

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

使用完全平方公式:

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

a = u^{2}, b = 1

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

化简

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

(u^{2})^{2} + 2 u^{2} \cdot 1 + 1^{2}

使用法则

1^{a} = 1

1^{2} = 1

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

2 u^{2} \cdot 1 = 2 u^{2}

2 u^{2} \cdot 1

数字相乘:

2 \cdot 1 = 2

= 2 u^{2}

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

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

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

乘开

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

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

打开括号

= 2 u^{4} + 2 \cdot 2 u^{2} + 2 \cdot 1

化简

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

2 u^{4} + 2 \cdot 2 u^{2} + 2 \cdot 1

数字相乘:

2 \cdot 2 = 4

= 2 u^{4} + 4 u^{2} + 2 \cdot 1

数字相乘:

2 \cdot 1 = 2

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

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

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

化简

2 u^{4} + 4 u^{2} + 2 + u^{2} : 2 u^{4} + 5 u^{2} + 2

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

对同类项分组

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

同类项相加:

4 u^{2} + u^{2} = 5 u^{2}

= 2 u^{4} + 5 u^{2} + 2

= 2 u^{4} + 5 u^{2} + 2

2 u^{4} + 5 u^{2} + 2 = 0

用

v = u^{2}

和

v^{2} = u^{4}

改写方程式

2 v^{2} + 5 v + 2 = 0

解

2 v^{2} + 5 v + 2 = 0 : v = - \frac{1}{2}, v = - 2

2 v^{2} + 5 v + 2 = 0

使用求根公式求解

2 v^{2} + 5 v + 2 = 0

二次方程求根公式:

若

a = 2, b = 5, c = 2

v_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

v_{1, 2} = \frac{- 5 \pm 5 ^{2} - 4 \cdot 2 \cdot 2}{2 \cdot 2}

5^{2} - 4 \cdot 2 \cdot 2 = 3

5^{2} - 4 \cdot 2 \cdot 2

数字相乘:

4 \cdot 2 \cdot 2 = 16

= 5^{2} - 16

5^{2} = 25

= 25 - 16

数字相减:

25 - 16 = 9

= 9

因式分解数字:

9 = 3^{2}

= 3^{2}

使用根式运算法则:

3^{2} = 3

= 3

v_{1, 2} = \frac{- 5 \pm 3}{2 \cdot 2}

将解分隔开

v_{1} = \frac{- 5 + 3}{2 \cdot 2}, v_{2} = \frac{- 5 - 3}{2 \cdot 2}

v = \frac{- 5 + 3}{2 \cdot 2} : - \frac{1}{2}

\frac{- 5 + 3}{2 \cdot 2}

数字相加/相减:

- 5 + 3 = - 2

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

数字相乘:

2 \cdot 2 = 4

= \frac{- 2}{4}

使用分式法则:

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

= - \frac{2}{4}

约分:

2

= - \frac{1}{2}

v = \frac{- 5 - 3}{2 \cdot 2} : - 2

\frac{- 5 - 3}{2 \cdot 2}

数字相减:

- 5 - 3 = - 8

= \frac{- 8}{2 \cdot 2}

数字相乘:

2 \cdot 2 = 4

= \frac{- 8}{4}

使用分式法则:

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

= - \frac{8}{4}

数字相除:

\frac{8}{4} = 2

= - 2

二次方程组的解是:

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

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

代回

v = u^{2}

，求解

u

解

u^{2} = - \frac{1}{2} : u \in R

无解

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

x^{2}

在

x

内不能为负

\in R

u \in R 无解

解

u^{2} = - 2 : u \in R

无解

u^{2} = - 2

x^{2}

在

x

内不能为负

\in R

u \in R 无解

解是

u \in R 无解

u \in R 无解

解

2 (u + \frac{1}{u}) + 3 = 0 : u = - \frac{1}{2}, u = - 2

2 (u + \frac{1}{u}) + 3 = 0

展开

2 (u + \frac{1}{u}) + 3 : 2 u + \frac{2}{u} + 3

2 (u + \frac{1}{u}) + 3

乘开

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

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

使用分配律:

a (b + c) = ab + a c

a = 2, b = u, c = \frac{1}{u}

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

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

2 \frac{1}{u}

分式相乘:

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

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

乘以:

1 \cdot 2 = 2

= \frac{2}{u}

= 2 u + \frac{2}{u}

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

2 u + \frac{2}{u} + 3 = 0

在两边乘以

u

2 u + \frac{2}{u} + 3 = 0

在两边乘以

u

2 uu + \frac{2}{u} u + 3 u = 0 \cdot u

化简

2 uu + \frac{2}{u} u + 3 u = 0 \cdot u

化简

2 uu : 2 u^{2}

2 uu

使用指数法则:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

数字相加:

1 + 1 = 2

= 2 u^{2}

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

2 u^{2} + 2 + 3 u = 0

2 u^{2} + 2 + 3 u = 0

2 u^{2} + 2 + 3 u = 0

解

2 u^{2} + 2 + 3 u = 0 : u = - \frac{1}{2}, u = - 2

2 u^{2} + 2 + 3 u = 0

改写成标准形式

a x^{2} + b x + c = 0

2 u^{2} + 3 u + 2 = 0

使用求根公式求解

2 u^{2} + 3 u + 2 = 0

二次方程求根公式:

若

a = 2, b = 3, c = 2

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 2 2}{2 2}

u_{1, 2} = \frac{- 3 \pm 3 ^{2} - 4 2 2}{2 2}

3^{2} - 422 = 1

3^{2} - 422

422 = 8

422

使用根式运算法则:

a a = a

22 = 2

= 4 \cdot 2

数字相乘:

4 \cdot 2 = 8

= 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数字相减:

9 - 8 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- 3 \pm 1}{2 2}

将解分隔开

u_{1} = \frac{- 3 + 1}{2 2}, u_{2} = \frac{- 3 - 1}{2 2}

u = \frac{- 3 + 1}{2 2} : - \frac{1}{2}

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

数字相加/相减:

- 3 + 1 = - 2

= \frac{- 2}{2 2}

使用分式法则:

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

= - \frac{2}{2 2}

数字相除:

\frac{2}{2} = 1

= - \frac{1}{2}

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

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

数字相减:

- 3 - 1 = - 4

= \frac{- 4}{2 2}

使用分式法则:

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

= - \frac{4}{2 2}

数字相除:

\frac{4}{2} = 2

= \frac{2}{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

数字相减:

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

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

使用根式运算法则:

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

= - 2

二次方程组的解是:

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

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

验证解

找到无定义的点（奇点）:

u = 0

取

2 (u + \frac{1}{u}) + 3

的分母，令其等于零

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}{2}, u = - 2

解

2 (u + \frac{1}{u}) - 3 = 0 : u = 2, u = \frac{1}{2}

2 (u + \frac{1}{u}) - 3 = 0

展开

2 (u + \frac{1}{u}) - 3 : 2 u + \frac{2}{u} - 3

2 (u + \frac{1}{u}) - 3

乘开

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

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

使用分配律:

a (b + c) = ab + a c

a = 2, b = u, c = \frac{1}{u}

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

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

2 \frac{1}{u}

分式相乘:

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

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

乘以:

1 \cdot 2 = 2

= \frac{2}{u}

= 2 u + \frac{2}{u}

= 2 u + \frac{2}{u} - 3

2 u + \frac{2}{u} - 3 = 0

在两边乘以

u

2 u + \frac{2}{u} - 3 = 0

在两边乘以

u

2 uu + \frac{2}{u} u - 3 u = 0 \cdot u

化简

2 uu + \frac{2}{u} u - 3 u = 0 \cdot u

化简

2 uu : 2 u^{2}

2 uu

使用指数法则:

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

uu = u^{1 + 1}

= 2 u^{1 + 1}

数字相加:

1 + 1 = 2

= 2 u^{2}

化简

\frac{2}{u} u : 2

\frac{2}{u} u

分式相乘:

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

= \frac{2 u}{u}

约分:

u

= 2

化简

0 \cdot u : 0

0 \cdot u

使用法则

0 \cdot a = 0

= 0

2 u^{2} + 2 - 3 u = 0

2 u^{2} + 2 - 3 u = 0

2 u^{2} + 2 - 3 u = 0

解

2 u^{2} + 2 - 3 u = 0 : u = 2, u = \frac{1}{2}

2 u^{2} + 2 - 3 u = 0

改写成标准形式

a x^{2} + b x + c = 0

2 u^{2} - 3 u + 2 = 0

使用求根公式求解

2 u^{2} - 3 u + 2 = 0

二次方程求根公式:

若

a = 2, b = - 3, c = 2

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 2 2}{2 2}

u_{1, 2} = \frac{- ( - 3 ) \pm ( - 3 ) ^{2} - 4 2 2}{2 2}

(- 3)^{2} - 422 = 1

(- 3)^{2} - 422

(- 3)^{2} = 3^{2}

(- 3)^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- 3)^{2} = 3^{2}

= 3^{2}

422 = 8

422

使用根式运算法则:

a a = a

22 = 2

= 4 \cdot 2

数字相乘:

4 \cdot 2 = 8

= 8

= 3^{2} - 8

3^{2} = 9

= 9 - 8

数字相减:

9 - 8 = 1

= 1

使用法则

1 = 1

= 1

u_{1, 2} = \frac{- ( - 3 ) \pm 1}{2 2}

将解分隔开

u_{1} = \frac{- ( - 3 ) + 1}{2 2}, u_{2} = \frac{- ( - 3 ) - 1}{2 2}

u = \frac{- ( - 3 ) + 1}{2 2} : 2

\frac{- ( - 3 ) + 1}{2 2}

使用法则

- (- a) = a

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

数字相加:

3 + 1 = 4

= \frac{4}{2 2}

数字相除:

\frac{4}{2} = 2

= \frac{2}{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

数字相减:

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

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

使用根式运算法则:

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

= 2

u = \frac{- ( - 3 ) - 1}{2 2} : \frac{1}{2}

\frac{- ( - 3 ) - 1}{2 2}

使用法则

- (- a) = a

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

数字相减:

3 - 1 = 2

= \frac{2}{2 2}

数字相除:

\frac{2}{2} = 1

= \frac{1}{2}

二次方程组的解是:

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

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

验证解

找到无定义的点（奇点）:

u = 0

取

2 (u + \frac{1}{u}) - 3

的分母，令其等于零

u = 0

以下点无定义

u = 0

将不在定义域的点与解相综合:

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

验证解:

u = - \frac{1}{2}

真

, u = - 2

真

, u = 2

真

, u = \frac{1}{2}

真

将它们代入

64 (\frac{u + u ^{- 1}}{2})^{4} - 64 (\frac{u + u ^{- 1}}{2})^{2} - 9 = 0

检验解是否符合

去除与方程不符的解。

代入

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

真

64 \frac{( - \frac{1}{2} ) + ( - \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{( - \frac{1}{2} ) + ( - \frac{1}{2} ) ^{- 1}}{2}^{2} - 9 = 0

64 \frac{( - \frac{1}{2} ) + ( - \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{( - \frac{1}{2} ) + ( - \frac{1}{2} ) ^{- 1}}{2}^{2} - 9 = 0

64 \frac{( - \frac{1}{2} ) + ( - \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{( - \frac{1}{2} ) + ( - \frac{1}{2} ) ^{- 1}}{2}^{2} - 9

去除括号:

(- a) = - a

= 64 \frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2}^{2} - 9

64 \frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2}^{4} = 81

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

\frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2}^{4} = \frac{3 ^{4}}{2 ^{6}}

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

\frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2} = - \frac{3}{2 2}

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

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

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

使用指数法则:

(- a)^{n} = - a^{n},

若

n

是奇数

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

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

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

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

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

使用指数法则:

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

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

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{1}

使用分式法则:

\frac{a}{1} = a

= 2

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

数字相减:

- 1 - 2 = - 3

= - 3

= \frac{- 3}{2}

使用分式法则:

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

= - \frac{3}{2}

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

使用分式法则:

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

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

使用分式法则:

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

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

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

= (- \frac{3}{2 2})^{4}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3}{2 2})^{4} = (\frac{3}{2 \cdot 2})^{4}

= (\frac{3}{2 \cdot 2})^{4}

使用指数法则:

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

= \frac{3 ^{4}}{( 2 2 ) ^{4}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{4} = 2^{4} (2)^{4}

= \frac{3 ^{4}}{2 ^{4} ( 2 ) ^{4}}

(2)^{4} : 2^{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 4

分式相乘:

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

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

数字相乘:

1 \cdot 4 = 4

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

= 2^{2}

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

2^{2} \cdot 2^{4} = 2^{6}

2^{2} \cdot 2^{4}

使用指数法则:

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

2^{2} \cdot 2^{4} = 2^{2 + 4}

= 2^{2 + 4}

数字相加:

2 + 4 = 6

= 2^{6}

= \frac{3 ^{4}}{2 ^{6}}

= 64 \cdot \frac{3 ^{4}}{2 ^{6}}

分式相乘:

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

= \frac{3 ^{4} \cdot 64}{2 ^{6}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{4}}{2 ^{6}}

约分:

2^{6}

= 3^{4}

3^{4} = 81

= 81

64 \frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2}^{2} = 72

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

\frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2}^{2} = \frac{3 ^{2}}{2 ^{3}}

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

\frac{- \frac{1}{2} + ( - \frac{1}{2} ) ^{- 1}}{2} = - \frac{3}{2 2}

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

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

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

使用指数法则:

(- a)^{n} = - a^{n},

若

n

是奇数

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

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

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

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

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

使用指数法则:

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

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

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{1}

使用分式法则:

\frac{a}{1} = a

= 2

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

数字相减:

- 1 - 2 = - 3

= - 3

= \frac{- 3}{2}

使用分式法则:

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

= - \frac{3}{2}

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

使用分式法则:

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

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

使用分式法则:

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

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

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

= (- \frac{3}{2 2})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3}{2 2})^{2} = (\frac{3}{2 \cdot 2})^{2}

= (\frac{3}{2 \cdot 2})^{2}

使用指数法则:

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

= \frac{3 ^{2}}{( 2 2 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{2} = 2^{2} (2)^{2}

= \frac{3 ^{2}}{2 ^{2} ( 2 ) ^{2}}

(2)^{2} : 2

使用根式运算法则:

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

= (2^{\frac{1}{2}})^{2}

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 2

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

2 \cdot 2^{2} = 2^{3}

2 \cdot 2^{2}

使用指数法则:

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

2 \cdot 2^{2} = 2^{1 + 2}

= 2^{1 + 2}

数字相加:

1 + 2 = 3

= 2^{3}

= \frac{3 ^{2}}{2 ^{3}}

= 64 \cdot \frac{3 ^{2}}{2 ^{3}}

分式相乘:

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

= \frac{3 ^{2} \cdot 64}{2 ^{3}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{2}}{2 ^{3}}

消掉

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}} : 2^{3} \cdot 3^{2}

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}}

使用指数法则:

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

\frac{2 ^{6}}{2 ^{3}} = 2^{6 - 3}

= 3^{2} \cdot 2^{6 - 3}

数字相减:

6 - 3 = 3

= 2^{3} \cdot 3^{2}

= 2^{3} \cdot 3^{2}

2^{3} = 8

= 3^{2} \cdot 8

3^{2} = 9

= 8 \cdot 9

数字相乘:

8 \cdot 9 = 72

= 72

= 81 - 72 - 9

数字相减:

81 - 72 - 9 = 0

= 0

0 = 0

真

代入

u = - 2 :

真

64 (\frac{( - 2 ) + ( - 2 ) ^{- 1}}{2})^{4} - 64 (\frac{( - 2 ) + ( - 2 ) ^{- 1}}{2})^{2} - 9 = 0

64 (\frac{( - 2 ) + ( - 2 ) ^{- 1}}{2})^{4} - 64 (\frac{( - 2 ) + ( - 2 ) ^{- 1}}{2})^{2} - 9 = 0

64 (\frac{( - 2 ) + ( - 2 ) ^{- 1}}{2})^{4} - 64 (\frac{( - 2 ) + ( - 2 ) ^{- 1}}{2})^{2} - 9

去除括号:

(- a) = - a

= 64 (\frac{- 2 + ( - 2 ) ^{- 1}}{2})^{4} - 64 (\frac{- 2 + ( - 2 ) ^{- 1}}{2})^{2} - 9

64 (\frac{- 2 + ( - 2 ) ^{- 1}}{2})^{4} = 81

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

(\frac{- 2 + ( - 2 ) ^{- 1}}{2})^{4} = \frac{3 ^{4}}{2 ^{6}}

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

\frac{- 2 + ( - 2 ) ^{- 1}}{2} = - \frac{3}{2 2}

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

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

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

使用指数法则:

(- a)^{n} = - a^{n},

若

n

是奇数

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

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

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

使用指数法则:

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

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

化简

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

- 2 - \frac{1}{2}

将项转换为分式:

2 = \frac{2 2}{2}

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

- 22 - 1 = - 3

- 22 - 1

使用根式运算法则:

a a = a

22 = 2

= - 2 - 1

数字相减:

- 2 - 1 = - 3

= - 3

= \frac{- 3}{2}

使用分式法则:

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

= - \frac{3}{2}

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

使用分式法则:

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

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

使用分式法则:

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

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

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

= (- \frac{3}{2 2})^{4}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3}{2 2})^{4} = (\frac{3}{2 \cdot 2})^{4}

= (\frac{3}{2 \cdot 2})^{4}

使用指数法则:

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

= \frac{3 ^{4}}{( 2 2 ) ^{4}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{4} = 2^{4} (2)^{4}

= \frac{3 ^{4}}{2 ^{4} ( 2 ) ^{4}}

(2)^{4} : 2^{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 4

分式相乘:

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

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

数字相乘:

1 \cdot 4 = 4

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

= 2^{2}

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

2^{2} \cdot 2^{4} = 2^{6}

2^{2} \cdot 2^{4}

使用指数法则:

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

2^{2} \cdot 2^{4} = 2^{2 + 4}

= 2^{2 + 4}

数字相加:

2 + 4 = 6

= 2^{6}

= \frac{3 ^{4}}{2 ^{6}}

= 64 \cdot \frac{3 ^{4}}{2 ^{6}}

分式相乘:

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

= \frac{3 ^{4} \cdot 64}{2 ^{6}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{4}}{2 ^{6}}

约分:

2^{6}

= 3^{4}

3^{4} = 81

= 81

64 (\frac{- 2 + ( - 2 ) ^{- 1}}{2})^{2} = 72

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

(\frac{- 2 + ( - 2 ) ^{- 1}}{2})^{2} = \frac{3 ^{2}}{2 ^{3}}

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

\frac{- 2 + ( - 2 ) ^{- 1}}{2} = - \frac{3}{2 2}

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

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

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

使用指数法则:

(- a)^{n} = - a^{n},

若

n

是奇数

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

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

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

使用指数法则:

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

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

化简

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

- 2 - \frac{1}{2}

将项转换为分式:

2 = \frac{2 2}{2}

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

- 22 - 1 = - 3

- 22 - 1

使用根式运算法则:

a a = a

22 = 2

= - 2 - 1

数字相减:

- 2 - 1 = - 3

= - 3

= \frac{- 3}{2}

使用分式法则:

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

= - \frac{3}{2}

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

使用分式法则:

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

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

使用分式法则:

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

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

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

= (- \frac{3}{2 2})^{2}

使用指数法则:

(- a)^{n} = a^{n},

若

n

是偶数

(- \frac{3}{2 2})^{2} = (\frac{3}{2 \cdot 2})^{2}

= (\frac{3}{2 \cdot 2})^{2}

使用指数法则:

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

= \frac{3 ^{2}}{( 2 2 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{2} = 2^{2} (2)^{2}

= \frac{3 ^{2}}{2 ^{2} ( 2 ) ^{2}}

(2)^{2} : 2

使用根式运算法则:

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

= (2^{\frac{1}{2}})^{2}

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 2

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

2 \cdot 2^{2} = 2^{3}

2 \cdot 2^{2}

使用指数法则:

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

2 \cdot 2^{2} = 2^{1 + 2}

= 2^{1 + 2}

数字相加:

1 + 2 = 3

= 2^{3}

= \frac{3 ^{2}}{2 ^{3}}

= 64 \cdot \frac{3 ^{2}}{2 ^{3}}

分式相乘:

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

= \frac{3 ^{2} \cdot 64}{2 ^{3}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{2}}{2 ^{3}}

消掉

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}} : 2^{3} \cdot 3^{2}

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}}

使用指数法则:

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

\frac{2 ^{6}}{2 ^{3}} = 2^{6 - 3}

= 3^{2} \cdot 2^{6 - 3}

数字相减:

6 - 3 = 3

= 2^{3} \cdot 3^{2}

= 2^{3} \cdot 3^{2}

2^{3} = 8

= 3^{2} \cdot 8

3^{2} = 9

= 8 \cdot 9

数字相乘:

8 \cdot 9 = 72

= 72

= 81 - 72 - 9

数字相减:

81 - 72 - 9 = 0

= 0

0 = 0

真

代入

u = 2 :

真

64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{4} - 64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{2} - 9 = 0

64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{4} - 64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{2} - 9 = 0

64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{4} - 64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{2} - 9

64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{4} = 81

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

(\frac{2 + ( 2 ) ^{- 1}}{2})^{4} = \frac{3 ^{4}}{2 ^{6}}

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

\frac{2 + ( 2 ) ^{- 1}}{2} = \frac{3}{2 2}

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

使用指数法则:

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

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

化简

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

2 + \frac{1}{2}

将项转换为分式:

2 = \frac{2 2}{2}

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

22 + 1 = 3

22 + 1

使用根式运算法则:

a a = a

22 = 2

= 2 + 1

数字相加:

2 + 1 = 3

= 3

= \frac{3}{2}

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

使用分式法则:

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

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

= (\frac{3}{2 \cdot 2})^{4}

使用指数法则:

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

= \frac{3 ^{4}}{( 2 2 ) ^{4}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{4} = 2^{4} (2)^{4}

= \frac{3 ^{4}}{2 ^{4} ( 2 ) ^{4}}

(2)^{4} : 2^{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 4

分式相乘:

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

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

数字相乘:

1 \cdot 4 = 4

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

= 2^{2}

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

2^{2} \cdot 2^{4} = 2^{6}

2^{2} \cdot 2^{4}

使用指数法则:

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

2^{2} \cdot 2^{4} = 2^{2 + 4}

= 2^{2 + 4}

数字相加:

2 + 4 = 6

= 2^{6}

= \frac{3 ^{4}}{2 ^{6}}

= 64 \cdot \frac{3 ^{4}}{2 ^{6}}

分式相乘:

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

= \frac{3 ^{4} \cdot 64}{2 ^{6}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{4}}{2 ^{6}}

约分:

2^{6}

= 3^{4}

3^{4} = 81

= 81

64 (\frac{2 + ( 2 ) ^{- 1}}{2})^{2} = 72

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

(\frac{2 + ( 2 ) ^{- 1}}{2})^{2} = \frac{3 ^{2}}{2 ^{3}}

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

\frac{2 + ( 2 ) ^{- 1}}{2} = \frac{3}{2 2}

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

使用指数法则:

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

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

化简

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

2 + \frac{1}{2}

将项转换为分式:

2 = \frac{2 2}{2}

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

22 + 1 = 3

22 + 1

使用根式运算法则:

a a = a

22 = 2

= 2 + 1

数字相加:

2 + 1 = 3

= 3

= \frac{3}{2}

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

使用分式法则:

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

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

= (\frac{3}{2 \cdot 2})^{2}

使用指数法则:

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

= \frac{3 ^{2}}{( 2 2 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{2} = 2^{2} (2)^{2}

= \frac{3 ^{2}}{2 ^{2} ( 2 ) ^{2}}

(2)^{2} : 2

使用根式运算法则:

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

= (2^{\frac{1}{2}})^{2}

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 2

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

2 \cdot 2^{2} = 2^{3}

2 \cdot 2^{2}

使用指数法则:

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

2 \cdot 2^{2} = 2^{1 + 2}

= 2^{1 + 2}

数字相加:

1 + 2 = 3

= 2^{3}

= \frac{3 ^{2}}{2 ^{3}}

= 64 \cdot \frac{3 ^{2}}{2 ^{3}}

分式相乘:

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

= \frac{3 ^{2} \cdot 64}{2 ^{3}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{2}}{2 ^{3}}

消掉

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}} : 2^{3} \cdot 3^{2}

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}}

使用指数法则:

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

\frac{2 ^{6}}{2 ^{3}} = 2^{6 - 3}

= 3^{2} \cdot 2^{6 - 3}

数字相减:

6 - 3 = 3

= 2^{3} \cdot 3^{2}

= 2^{3} \cdot 3^{2}

2^{3} = 8

= 3^{2} \cdot 8

3^{2} = 9

= 8 \cdot 9

数字相乘:

8 \cdot 9 = 72

= 72

= 81 - 72 - 9

数字相减:

81 - 72 - 9 = 0

= 0

0 = 0

真

代入

u = \frac{1}{2} :

真

64 \frac{( \frac{1}{2} ) + ( \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{( \frac{1}{2} ) + ( \frac{1}{2} ) ^{- 1}}{2}^{2} - 9 = 0

64 \frac{( \frac{1}{2} ) + ( \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{( \frac{1}{2} ) + ( \frac{1}{2} ) ^{- 1}}{2}^{2} - 9 = 0

64 \frac{( \frac{1}{2} ) + ( \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{( \frac{1}{2} ) + ( \frac{1}{2} ) ^{- 1}}{2}^{2} - 9

去除括号:

(a) = a

= 64 \frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2}^{4} - 64 \frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2}^{2} - 9

64 \frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2}^{4} = 81

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

\frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2}^{4} = \frac{3 ^{4}}{2 ^{6}}

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

\frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2} = \frac{3}{2 2}

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

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

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

使用指数法则:

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

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

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{1}

使用分式法则:

\frac{a}{1} = a

= 2

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

使用分式法则:

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

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

= (\frac{3}{2 \cdot 2})^{4}

使用指数法则:

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

= \frac{3 ^{4}}{( 2 2 ) ^{4}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{4} = 2^{4} (2)^{4}

= \frac{3 ^{4}}{2 ^{4} ( 2 ) ^{4}}

(2)^{4} : 2^{2}

使用根式运算法则:

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

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

使用指数法则:

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

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

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

\frac{1}{2} \cdot 4

分式相乘:

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

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

数字相乘:

1 \cdot 4 = 4

= \frac{4}{2}

数字相除:

\frac{4}{2} = 2

= 2

= 2^{2}

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

2^{2} \cdot 2^{4} = 2^{6}

2^{2} \cdot 2^{4}

使用指数法则:

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

2^{2} \cdot 2^{4} = 2^{2 + 4}

= 2^{2 + 4}

数字相加:

2 + 4 = 6

= 2^{6}

= \frac{3 ^{4}}{2 ^{6}}

= 64 \cdot \frac{3 ^{4}}{2 ^{6}}

分式相乘:

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

= \frac{3 ^{4} \cdot 64}{2 ^{6}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{4}}{2 ^{6}}

约分:

2^{6}

= 3^{4}

3^{4} = 81

= 81

64 \frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2}^{2} = 72

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

\frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2}^{2} = \frac{3 ^{2}}{2 ^{3}}

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

\frac{\frac{1}{2} + ( \frac{1}{2} ) ^{- 1}}{2} = \frac{3}{2 2}

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

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

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

使用指数法则:

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

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

使用分式法则:

\frac{1}{\frac{b}{c}} = \frac{c}{b}

= \frac{2}{1}

使用分式法则:

\frac{a}{1} = a

= 2

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

使用分式法则:

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

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

= (\frac{3}{2 \cdot 2})^{2}

使用指数法则:

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

= \frac{3 ^{2}}{( 2 2 ) ^{2}}

使用指数法则:

(a \cdot b)^{n} = a^{n} b^{n}

(22)^{2} = 2^{2} (2)^{2}

= \frac{3 ^{2}}{2 ^{2} ( 2 ) ^{2}}

(2)^{2} : 2

使用根式运算法则:

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

= (2^{\frac{1}{2}})^{2}

使用指数法则:

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

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

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

\frac{1}{2} \cdot 2

分式相乘:

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

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

约分:

2

= 1

= 2

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

2 \cdot 2^{2} = 2^{3}

2 \cdot 2^{2}

使用指数法则:

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

2 \cdot 2^{2} = 2^{1 + 2}

= 2^{1 + 2}

数字相加:

1 + 2 = 3

= 2^{3}

= \frac{3 ^{2}}{2 ^{3}}

= 64 \cdot \frac{3 ^{2}}{2 ^{3}}

分式相乘:

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

= \frac{3 ^{2} \cdot 64}{2 ^{3}}

分解

64 : 2^{6}

因式分解

64 = 2^{6}

= \frac{2 ^{6} \cdot 3 ^{2}}{2 ^{3}}

消掉

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}} : 2^{3} \cdot 3^{2}

\frac{3 ^{2} \cdot 2 ^{6}}{2 ^{3}}

使用指数法则:

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

\frac{2 ^{6}}{2 ^{3}} = 2^{6 - 3}

= 3^{2} \cdot 2^{6 - 3}

数字相减:

6 - 3 = 3

= 2^{3} \cdot 3^{2}

= 2^{3} \cdot 3^{2}

2^{3} = 8

= 3^{2} \cdot 8

3^{2} = 9

= 8 \cdot 9

数字相乘:

8 \cdot 9 = 72

= 72

= 81 - 72 - 9

数字相减:

81 - 72 - 9 = 0

= 0

0 = 0

真

解为

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

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

代回

u = e^{x}

，求解

x

解

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

无解

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

使用指数运算法则

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

使用指数法则:

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

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

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

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

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

解

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

无解

e^{x} = - 2

a^{f (x)}

对于

x

不能为零或负值

\in R

x \in R 无解

解

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

e^{x} = 2

使用指数运算法则

e^{x} = 2

使用指数法则:

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

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

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

若

f (x) = g (x)

，则

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

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

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

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

使用对数计算法则:

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

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

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

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

解

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

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

使用指数运算法则

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

使用指数法则:

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

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

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

使用指数法则:

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

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

若

f (x) = g (x)

，则

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

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

使用对数计算法则:

ln (e^{a}) = a

ln (e^{x}) = x

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

使用对数计算法则:

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

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

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

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

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

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

流行的例子

1 + cos (θ) = 1

tan (2 θ) = 0.4

csc(θ/2)=sin(θ/2)

csc (\frac{θ}{2}) = sin (\frac{θ}{2})

sin(3x)=-1,0<= x<= 2pi

sin (3 x) = - 1, 0 \leq x \leq 2 π

sin (θ) = 0.25