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受欢迎的三角函数 >

cos^4(x)+2sin^2(x)+6cos^2(x)+5=0

解答

cos^{4} (x) + 2 sin^{2} (x) + 6 cos^{2} (x) + 5 = 0

解答

x \in R 无解

求解步骤

cos^{4} (x) + 2 sin^{2} (x) + 6 cos^{2} (x) + 5 = 0

使用三角恒等式改写

5 + cos^{4} (x) + 2 sin^{2} (x) + 6 cos^{2} (x)

使用毕达哥拉斯恒等式:

cos^{2} (x) + sin^{2} (x) = 1

sin^{2} (x) = 1 - cos^{2} (x)

= 5 + cos^{4} (x) + 2 (1 - cos^{2} (x)) + 6 cos^{2} (x)

化简

5 + cos^{4} (x) + 2 (1 - cos^{2} (x)) + 6 cos^{2} (x) : cos^{4} (x) + 4 cos^{2} (x) + 7

5 + cos^{4} (x) + 2 (1 - cos^{2} (x)) + 6 cos^{2} (x)

乘开

2 (1 - cos^{2} (x)) : 2 - 2 cos^{2} (x)

2 (1 - cos^{2} (x))

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = cos^{2} (x)

= 2 \cdot 1 - 2 cos^{2} (x)

数字相乘:

2 \cdot 1 = 2

= 2 - 2 cos^{2} (x)

= 5 + cos^{4} (x) + 2 - 2 cos^{2} (x) + 6 cos^{2} (x)

化简

5 + cos^{4} (x) + 2 - 2 cos^{2} (x) + 6 cos^{2} (x) : cos^{4} (x) + 4 cos^{2} (x) + 7

5 + cos^{4} (x) + 2 - 2 cos^{2} (x) + 6 cos^{2} (x)

同类项相加:

- 2 cos^{2} (x) + 6 cos^{2} (x) = 4 cos^{2} (x)

= 5 + cos^{4} (x) + 2 + 4 cos^{2} (x)

对同类项分组

= cos^{4} (x) + 4 cos^{2} (x) + 5 + 2

数字相加:

5 + 2 = 7

= cos^{4} (x) + 4 cos^{2} (x) + 7

= cos^{4} (x) + 4 cos^{2} (x) + 7

= cos^{4} (x) + 4 cos^{2} (x) + 7

7 + cos^{4} (x) + 4 cos^{2} (x) = 0

用替代法求解

7 + cos^{4} (x) + 4 cos^{2} (x) = 0

令

: cos (x) = u

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

7 + u^{4} + 4 u^{2} = 0 : u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

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

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

u^{4} + 4 u^{2} + 7 = 0

用

a = u^{2}

和

a^{2} = u^{4}

改写方程式

a^{2} + 4 a + 7 = 0

解

a^{2} + 4 a + 7 = 0 : a = - 2 + 3 i, a = - 2 - 3 i

a^{2} + 4 a + 7 = 0

使用求根公式求解

a^{2} + 4 a + 7 = 0

二次方程求根公式:

若

a = 1, b = 4, c = 7

a_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot 7}{2 \cdot 1}

a_{1, 2} = \frac{- 4 \pm 4 ^{2} - 4 \cdot 1 \cdot 7}{2 \cdot 1}

化简

4^{2} - 4 \cdot 1 \cdot 7 : 23 i

4^{2} - 4 \cdot 1 \cdot 7

数字相乘:

4 \cdot 1 \cdot 7 = 28

= 4^{2} - 28

使用虚数运算法则:

- a = i a

= i 28 - 4^{2}

- 4^{2} + 28 = 23

- 4^{2} + 28

4^{2} = 16

= - 16 + 28

数字相加/相减:

- 16 + 28 = 12

= 12

12

质因数分解:

2^{2} \cdot 3

12

12

除以

212 = 6 \cdot 2

= 2 \cdot 6

6

除以

26 = 3 \cdot 2

= 2 \cdot 2 \cdot 3

2, 3

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 3

= 2^{2} \cdot 3

= 2^{2} \cdot 3

使用根式运算法则:

n ab = n a n b

= 3 2^{2}

使用根式运算法则:

n a^{n} = a

2^{2} = 2

= 23

= 23 i

a_{1, 2} = \frac{- 4 \pm 2 3 i}{2 \cdot 1}

将解分隔开

a_{1} = \frac{- 4 + 2 3 i}{2 \cdot 1}, a_{2} = \frac{- 4 - 2 3 i}{2 \cdot 1}

a = \frac{- 4 + 2 3 i}{2 \cdot 1} : - 2 + 3 i

\frac{- 4 + 2 3 i}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

= \frac{- 4 + 2 3 i}{2}

分解

- 4 + 23 i : 2 (- 2 + 3 i)

- 4 + 23 i

改写为

= - 2 \cdot 2 + 23 i

因式分解出通项

2

= 2 (- 2 + 3 i)

= \frac{2 ( - 2 + 3 i )}{2}

数字相除:

\frac{2}{2} = 1

= - 2 + 3 i

a = \frac{- 4 - 2 3 i}{2 \cdot 1} : - 2 - 3 i

\frac{- 4 - 2 3 i}{2 \cdot 1}

数字相乘:

2 \cdot 1 = 2

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

分解

- 4 - 23 i : - 2 (2 + 3 i)

- 4 - 23 i

改写为

= - 2 \cdot 2 - 23 i

因式分解出通项

2

= - 2 (2 + 3 i)

= - \frac{2 ( 2 + 3 i )}{2}

数字相除:

\frac{2}{2} = 1

= - (2 + 3 i)

取负

- (2 + 3 i) = - 2 - 3 i

= - 2 - 3 i

二次方程组的解是:

a = - 2 + 3 i, a = - 2 - 3 i

a = - 2 + 3 i, a = - 2 - 3 i

代回

a = u^{2}

，求解

u

解

u^{2} = - 2 + 3 i : u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

u^{2} = - 2 + 3 i

替代

u = a + bi

(a + bi)^{2} = - 2 + 3 i

展开

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

使用完全平方公式:

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

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

使用指数法则:

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

= i^{2} b^{2}

i^{2} = - 1

i^{2}

使用虚数运算法则:

i^{2} = - 1

= - 1

= (- 1) b^{2}

整理后得

= - b^{2}

= a^{2} + 2 iab - b^{2}

将

a^{2} + 2 iab - b^{2}

改写成标准复数形式:

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

a^{2} + 2 iab - b^{2}

将复数的实部和虚部分组

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

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

(a^{2} - b^{2}) + 2 iab = - 2 + 3 i

复数仅在实部和虚部均相等时才相等改写为方程组:

[a^{2} - b^{2} = - 2 2 ab = 3]

[a^{2} - b^{2} = - 2 2 ab = 3] : a = \frac{3}{2 2 + 7}, a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

[a^{2} - b^{2} = - 2 2 ab = 3]

对于

2 ab = 3

将

a

移到一边:

a = \frac{3}{2 b}

2 ab = 3

两边除以

2 b

2 ab = 3

两边除以

2 b

\frac{2 ab}{2 b} = \frac{3}{2 b}

化简

a = \frac{3}{2 b}

a = \frac{3}{2 b}

将解

a = \frac{3}{2 b}

代入

a^{2} - b^{2} = - 2

对于

a^{2} - b^{2} = - 2

，用

\frac{3}{2 b}

替代

a : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

对于

a^{2} - b^{2} = - 2

，用

\frac{3}{2 b}

替代

a

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

解

(\frac{3}{2 b})^{2} - b^{2} = - 2 : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

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

化简

(\frac{3}{2 b})^{2} : \frac{3}{4 b ^{2}}

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

使用指数法则:

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

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

使用指数法则:

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

(2 b)^{2} = 2^{2} b^{2}

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

(3)^{2} : 3

使用根式运算法则:

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

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

使用指数法则:

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

= 3^{\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

= 3

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

2^{2} = 4

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

\frac{3}{4 b ^{2}} - b^{2} = - 2

在两边乘以

4 b^{2}

\frac{3}{4 b ^{2}} - b^{2} = - 2

在两边乘以

4 b^{2}

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

化简

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

化简

\frac{3}{4 b ^{2}} \cdot 4 b^{2} : 3

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

分式相乘:

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

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

约分:

4

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

约分:

b^{2}

= 3

化简

- b^{2} \cdot 4 b^{2} : - 4 b^{4}

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

使用指数法则:

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

b^{2} b^{2} = b^{2 + 2}

= - 4 b^{2 + 2}

数字相加:

2 + 2 = 4

= - 4 b^{4}

化简

- 2 \cdot 4 b^{2} : - 8 b^{2}

- 2 \cdot 4 b^{2}

数字相乘:

2 \cdot 4 = 8

= - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

解

3 - 4 b^{4} = - 8 b^{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

3 - 4 b^{4} = - 8 b^{2}

将

8 b^{2}

para o lado esquerdo

3 - 4 b^{4} = - 8 b^{2}

两边加上

8 b^{2}

3 - 4 b^{4} + 8 b^{2} = - 8 b^{2} + 8 b^{2}

化简

3 - 4 b^{4} + 8 b^{2} = 0

3 - 4 b^{4} + 8 b^{2} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 4 b^{4} + 8 b^{2} + 3 = 0

用

u = b^{2}

和

u^{2} = b^{4}

改写方程式

- 4 u^{2} + 8 u + 3 = 0

解

- 4 u^{2} + 8 u + 3 = 0 : u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

- 4 u^{2} + 8 u + 3 = 0

使用求根公式求解

- 4 u^{2} + 8 u + 3 = 0

二次方程求根公式:

若

a = - 4, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

8^{2} - 4 (- 4) \cdot 3 = 47

8^{2} - 4 (- 4) \cdot 3

使用法则

- (- a) = a

= 8^{2} + 4 \cdot 4 \cdot 3

数字相乘:

4 \cdot 4 \cdot 3 = 48

= 8^{2} + 48

8^{2} = 64

= 64 + 48

数字相加:

64 + 48 = 112

= 112

112

质因数分解:

2^{4} \cdot 7

112

112

除以

2112 = 56 \cdot 2

= 2 \cdot 56

56

除以

256 = 28 \cdot 2

= 2 \cdot 2 \cdot 28

28

除以

228 = 14 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

= 2^{4} \cdot 7

= 2^{4} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{4}

使用根式运算法则:

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

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

= 2^{2} 7

整理后得

= 47

u_{1, 2} = \frac{- 8 \pm 4 7}{2 ( - 4 )}

将解分隔开

u_{1} = \frac{- 8 + 4 7}{2 ( - 4 )}, u_{2} = \frac{- 8 - 4 7}{2 ( - 4 )}

u = \frac{- 8 + 4 7}{2 ( - 4 )} : - \frac{- 2 + 7}{2}

\frac{- 8 + 4 7}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 8 + 4 7}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 8 + 4 7}{- 8}

使用分式法则:

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

= - \frac{- 8 + 4 7}{8}

消掉

\frac{- 8 + 4 7}{8} : \frac{7 - 2}{2}

\frac{- 8 + 4 7}{8}

分解

- 8 + 47 : 4 (- 2 + 7)

- 8 + 47

改写为

= - 4 \cdot 2 + 47

因式分解出通项

4

= 4 (- 2 + 7)

= \frac{4 ( - 2 + 7 )}{8}

约分:

4

= \frac{- 2 + 7}{2}

= - \frac{7 - 2}{2}

= - \frac{- 2 + 7}{2}

u = \frac{- 8 - 4 7}{2 ( - 4 )} : \frac{2 + 7}{2}

\frac{- 8 - 4 7}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 8 - 4 7}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 8 - 4 7}{- 8}

使用分式法则:

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

- 8 - 47 = - (8 + 47)

= \frac{8 + 4 7}{8}

分解

8 + 47 : 4 (2 + 7)

8 + 47

改写为

= 4 \cdot 2 + 47

因式分解出通项

4

= 4 (2 + 7)

= \frac{4 ( 2 + 7 )}{8}

约分:

4

= \frac{2 + 7}{2}

二次方程组的解是:

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

代回

u = b^{2}

，求解

b

解

b^{2} = - \frac{- 2 + 7}{2} : b \in R

无解

b^{2} = - \frac{- 2 + 7}{2}

x^{2}

在

x

内不能为负

\in R

b \in R 无解

解

b^{2} = \frac{2 + 7}{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b^{2} = \frac{2 + 7}{2}

对于

x^{2} = f (a)

解为

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

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

解为

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

验证解

找到无定义的点（奇点）:

b = 0

取

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

的分母，令其等于零

解

2 b = 0 : b = 0

2 b = 0

两边除以

2

2 b = 0

两边除以

2

\frac{2 b}{2} = \frac{0}{2}

化简

b = 0

b = 0

以下点无定义

b = 0

将不在定义域的点与解相综合:

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

将解

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

代入

2 ab = 3

对于

2 ab = 3

，用

\frac{2 + 7}{2}

替代

b : a = \frac{3}{2 2 + 7}

对于

2 ab = 3

，用

\frac{2 + 7}{2}

替代

b

2 a \frac{2 + 7}{2} = 3

解

2 a \frac{2 + 7}{2} = 3 : a = \frac{3}{2 2 + 7}

2 a \frac{2 + 7}{2} = 3

两边除以

2 \frac{2 + 7}{2}

2 a \frac{2 + 7}{2} = 3

两边除以

2 \frac{2 + 7}{2}

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{3}{2 \frac{2 + 7}{2}}

化简

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{3}{2 \frac{2 + 7}{2}}

化简

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} : a

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

数字相除:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

约分:

\frac{2 + 7}{2}

= a

化简

\frac{3}{2 \frac{2 + 7}{2}} : \frac{3}{2 2 + 7}

\frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

使用根式运算法则:

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

假定

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= \frac{3}{2 \cdot \frac{2 + 7}{2}}

乘

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

分式相乘:

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

= \frac{2 + 7 \cdot 2}{2}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{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^{- \frac{1}{2} + 1} 2 + 7

数字相减:

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

= 2^{\frac{1}{2}} 2 + 7

使用根式运算法则:

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

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

= 2 2 + 7

= \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

对于

2 ab = 3

，用

- \frac{2 + 7}{2}

替代

b : a = - \frac{3}{2 2 + 7}

对于

2 ab = 3

，用

- \frac{2 + 7}{2}

替代

b

2 a - \frac{2 + 7}{2} = 3

解

2 a - \frac{2 + 7}{2} = 3 : a = - \frac{3}{2 2 + 7}

2 a - \frac{2 + 7}{2} = 3

两边除以

2 - \frac{2 + 7}{2}

2 a - \frac{2 + 7}{2} = 3

两边除以

2 - \frac{2 + 7}{2}

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{3}{2 ( - \frac{2 + 7}{2} )}

化简

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{3}{2 ( - \frac{2 + 7}{2} )}

化简

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} : a

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )}

去除括号:

(- a) = - a

= \frac{- 2 a \frac{2 + 7}{2}}{- 2 \frac{2 + 7}{2}}

使用分式法则:

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

= \frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

数字相除:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

约分:

\frac{2 + 7}{2}

= a

化简

\frac{3}{2 ( - \frac{2 + 7}{2} )} : - \frac{3}{2 2 + 7}

\frac{3}{2 ( - \frac{2 + 7}{2} )}

去除括号:

(- a) = - a

= \frac{3}{- 2 \frac{2 + 7}{2}}

使用分式法则:

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

= - \frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

使用根式运算法则:

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

假定

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= - \frac{3}{2 \cdot \frac{2 + 7}{2}}

乘

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

分式相乘:

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

= \frac{2 + 7 \cdot 2}{2}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{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^{- \frac{1}{2} + 1} 2 + 7

数字相减:

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

= 2^{\frac{1}{2}} 2 + 7

使用根式运算法则:

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

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

= 2 2 + 7

= - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

将解代入原方程进行验证

将它们代入

a^{2} - b^{2} = - 2

检验解是否符合

去除与方程不符的解。

检验

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

的解:

真

a^{2} - b^{2} = - 2

代入

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

(- \frac{3}{2 2 + 7})^{2} - - \frac{2 + 7}{2}^{2} = - 2

整理后得

- 2 = - 2

真

检验

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

的解:

真

a^{2} - b^{2} = - 2

代入

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

(\frac{3}{2 2 + 7})^{2} - \frac{2 + 7}{2}^{2} = - 2

整理后得

- 2 = - 2

真

将它们代入

2 ab = 3

检验解是否符合

去除与方程不符的解。

检验

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

的解:

真

2 ab = 3

代入

a = - \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

2 (- \frac{3}{2 2 + 7}) - \frac{2 + 7}{2} = 3

整理后得

3 = 3

真

检验

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

的解:

真

2 ab = 3

代入

a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

2 \cdot \frac{3}{2 2 + 7} \frac{2 + 7}{2} = 3

整理后得

3 = 3

真

因而，

a^{2} - b^{2} = - 2, 2 ab = 3

最后的解是

a = \frac{3}{2 2 + 7}, a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

u = a + bi

代回

u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

解

u^{2} = - 2 - 3 i : u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

u^{2} = - 2 - 3 i

替代

u = a + bi

(a + bi)^{2} = - 2 - 3 i

展开

(a + bi)^{2} : (a^{2} - b^{2}) + 2 iab

(a + bi)^{2}

= (a + ib)^{2}

使用完全平方公式:

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

a = a, b = bi

= a^{2} + 2 abi + (bi)^{2}

(bi)^{2} = - b^{2}

(bi)^{2}

使用指数法则:

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

= i^{2} b^{2}

i^{2} = - 1

i^{2}

使用虚数运算法则:

i^{2} = - 1

= - 1

= (- 1) b^{2}

整理后得

= - b^{2}

= a^{2} + 2 iab - b^{2}

将

a^{2} + 2 iab - b^{2}

改写成标准复数形式:

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

a^{2} + 2 iab - b^{2}

将复数的实部和虚部分组

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

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

(a^{2} - b^{2}) + 2 iab = - 2 - 3 i

复数仅在实部和虚部均相等时才相等改写为方程组:

[a^{2} - b^{2} = - 2 2 ab = - 3]

[a^{2} - b^{2} = - 2 2 ab = - 3] : a = - \frac{3}{2 2 + 7}, a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

[a^{2} - b^{2} = - 2 2 ab = - 3]

对于

2 ab = - 3

将

a

移到一边:

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

2 ab = - 3

两边除以

2 b

2 ab = - 3

两边除以

2 b

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

化简

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

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

将解

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

代入

a^{2} - b^{2} = - 2

对于

a^{2} - b^{2} = - 2

，用

- \frac{3}{2 b}

替代

a : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

对于

a^{2} - b^{2} = - 2

，用

- \frac{3}{2 b}

替代

a

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

解

(- \frac{3}{2 b})^{2} - b^{2} = - 2 : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

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

化简

(- \frac{3}{2 b})^{2} : \frac{3}{4 b ^{2}}

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

使用指数法则:

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

若

n

是偶数

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

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

使用指数法则:

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

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

使用指数法则:

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

(2 b)^{2} = 2^{2} b^{2}

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

(3)^{2} : 3

使用根式运算法则:

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

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

使用指数法则:

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

= 3^{\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

= 3

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

2^{2} = 4

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

\frac{3}{4 b ^{2}} - b^{2} = - 2

在两边乘以

4 b^{2}

\frac{3}{4 b ^{2}} - b^{2} = - 2

在两边乘以

4 b^{2}

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

化简

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

化简

\frac{3}{4 b ^{2}} \cdot 4 b^{2} : 3

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

分式相乘:

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

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

约分:

4

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

约分:

b^{2}

= 3

化简

- b^{2} \cdot 4 b^{2} : - 4 b^{4}

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

使用指数法则:

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

b^{2} b^{2} = b^{2 + 2}

= - 4 b^{2 + 2}

数字相加:

2 + 2 = 4

= - 4 b^{4}

化简

- 2 \cdot 4 b^{2} : - 8 b^{2}

- 2 \cdot 4 b^{2}

数字相乘:

2 \cdot 4 = 8

= - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

3 - 4 b^{4} = - 8 b^{2}

解

3 - 4 b^{4} = - 8 b^{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

3 - 4 b^{4} = - 8 b^{2}

将

8 b^{2}

para o lado esquerdo

3 - 4 b^{4} = - 8 b^{2}

两边加上

8 b^{2}

3 - 4 b^{4} + 8 b^{2} = - 8 b^{2} + 8 b^{2}

化简

3 - 4 b^{4} + 8 b^{2} = 0

3 - 4 b^{4} + 8 b^{2} = 0

改写成标准形式

a_{n} x^{n} + \dots + a_{1} x + a_{0} = 0

- 4 b^{4} + 8 b^{2} + 3 = 0

用

u = b^{2}

和

u^{2} = b^{4}

改写方程式

- 4 u^{2} + 8 u + 3 = 0

解

- 4 u^{2} + 8 u + 3 = 0 : u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

- 4 u^{2} + 8 u + 3 = 0

使用求根公式求解

- 4 u^{2} + 8 u + 3 = 0

二次方程求根公式:

若

a = - 4, b = 8, c = 3

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

u_{1, 2} = \frac{- 8 \pm 8 ^{2} - 4 ( - 4 ) \cdot 3}{2 ( - 4 )}

8^{2} - 4 (- 4) \cdot 3 = 47

8^{2} - 4 (- 4) \cdot 3

使用法则

- (- a) = a

= 8^{2} + 4 \cdot 4 \cdot 3

数字相乘:

4 \cdot 4 \cdot 3 = 48

= 8^{2} + 48

8^{2} = 64

= 64 + 48

数字相加:

64 + 48 = 112

= 112

112

质因数分解:

2^{4} \cdot 7

112

112

除以

2112 = 56 \cdot 2

= 2 \cdot 56

56

除以

256 = 28 \cdot 2

= 2 \cdot 2 \cdot 28

28

除以

228 = 14 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 14

14

除以

214 = 7 \cdot 2

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

2, 7

都是质数，因此无法进一步因数分解

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 7

= 2^{4} \cdot 7

= 2^{4} \cdot 7

使用根式运算法则:

n ab = n a n b

= 7 2^{4}

使用根式运算法则:

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

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

= 2^{2} 7

整理后得

= 47

u_{1, 2} = \frac{- 8 \pm 4 7}{2 ( - 4 )}

将解分隔开

u_{1} = \frac{- 8 + 4 7}{2 ( - 4 )}, u_{2} = \frac{- 8 - 4 7}{2 ( - 4 )}

u = \frac{- 8 + 4 7}{2 ( - 4 )} : - \frac{- 2 + 7}{2}

\frac{- 8 + 4 7}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 8 + 4 7}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 8 + 4 7}{- 8}

使用分式法则:

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

= - \frac{- 8 + 4 7}{8}

消掉

\frac{- 8 + 4 7}{8} : \frac{7 - 2}{2}

\frac{- 8 + 4 7}{8}

分解

- 8 + 47 : 4 (- 2 + 7)

- 8 + 47

改写为

= - 4 \cdot 2 + 47

因式分解出通项

4

= 4 (- 2 + 7)

= \frac{4 ( - 2 + 7 )}{8}

约分:

4

= \frac{- 2 + 7}{2}

= - \frac{7 - 2}{2}

= - \frac{- 2 + 7}{2}

u = \frac{- 8 - 4 7}{2 ( - 4 )} : \frac{2 + 7}{2}

\frac{- 8 - 4 7}{2 ( - 4 )}

去除括号:

(- a) = - a

= \frac{- 8 - 4 7}{- 2 \cdot 4}

数字相乘:

2 \cdot 4 = 8

= \frac{- 8 - 4 7}{- 8}

使用分式法则:

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

- 8 - 47 = - (8 + 47)

= \frac{8 + 4 7}{8}

分解

8 + 47 : 4 (2 + 7)

8 + 47

改写为

= 4 \cdot 2 + 47

因式分解出通项

4

= 4 (2 + 7)

= \frac{4 ( 2 + 7 )}{8}

约分:

4

= \frac{2 + 7}{2}

二次方程组的解是:

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

u = - \frac{- 2 + 7}{2}, u = \frac{2 + 7}{2}

代回

u = b^{2}

，求解

b

解

b^{2} = - \frac{- 2 + 7}{2} : b \in R

无解

b^{2} = - \frac{- 2 + 7}{2}

x^{2}

在

x

内不能为负

\in R

b \in R 无解

解

b^{2} = \frac{2 + 7}{2} : b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b^{2} = \frac{2 + 7}{2}

对于

x^{2} = f (a)

解为

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

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

解为

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

验证解

找到无定义的点（奇点）:

b = 0

取

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

的分母，令其等于零

解

2 b = 0 : b = 0

2 b = 0

两边除以

2

2 b = 0

两边除以

2

\frac{2 b}{2} = \frac{0}{2}

化简

b = 0

b = 0

以下点无定义

b = 0

将不在定义域的点与解相综合:

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

将解

b = \frac{2 + 7}{2}, b = - \frac{2 + 7}{2}

代入

2 ab = - 3

对于

2 ab = - 3

，用

\frac{2 + 7}{2}

替代

b : a = - \frac{3}{2 2 + 7}

对于

2 ab = - 3

，用

\frac{2 + 7}{2}

替代

b

2 a \frac{2 + 7}{2} = - 3

解

2 a \frac{2 + 7}{2} = - 3 : a = - \frac{3}{2 2 + 7}

2 a \frac{2 + 7}{2} = - 3

两边除以

2 \frac{2 + 7}{2}

2 a \frac{2 + 7}{2} = - 3

两边除以

2 \frac{2 + 7}{2}

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{- 3}{2 \frac{2 + 7}{2}}

化简

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} = \frac{- 3}{2 \frac{2 + 7}{2}}

化简

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}} : a

\frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

数字相除:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

约分:

\frac{2 + 7}{2}

= a

化简

\frac{- 3}{2 \frac{2 + 7}{2}} : - \frac{3}{2 2 + 7}

\frac{- 3}{2 \frac{2 + 7}{2}}

使用分式法则:

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

= - \frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

使用根式运算法则:

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

假定

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= - \frac{3}{2 \cdot \frac{2 + 7}{2}}

乘

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

分式相乘:

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

= \frac{2 + 7 \cdot 2}{2}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{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^{- \frac{1}{2} + 1} 2 + 7

数字相减:

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

= 2^{\frac{1}{2}} 2 + 7

使用根式运算法则:

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

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

= 2 2 + 7

= - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

a = - \frac{3}{2 2 + 7}

对于

2 ab = - 3

，用

- \frac{2 + 7}{2}

替代

b : a = \frac{3}{2 2 + 7}

对于

2 ab = - 3

，用

- \frac{2 + 7}{2}

替代

b

2 a - \frac{2 + 7}{2} = - 3

解

2 a - \frac{2 + 7}{2} = - 3 : a = \frac{3}{2 2 + 7}

2 a - \frac{2 + 7}{2} = - 3

两边除以

2 - \frac{2 + 7}{2}

2 a - \frac{2 + 7}{2} = - 3

两边除以

2 - \frac{2 + 7}{2}

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{- 3}{2 ( - \frac{2 + 7}{2} )}

化简

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} = \frac{- 3}{2 ( - \frac{2 + 7}{2} )}

化简

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )} : a

\frac{2 a ( - \frac{2 + 7}{2} )}{2 ( - \frac{2 + 7}{2} )}

去除括号:

(- a) = - a

= \frac{- 2 a \frac{2 + 7}{2}}{- 2 \frac{2 + 7}{2}}

使用分式法则:

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

= \frac{2 a \frac{2 + 7}{2}}{2 \frac{2 + 7}{2}}

数字相除:

\frac{2}{2} = 1

= \frac{\frac{2 + 7}{2} a}{\frac{2 + 7}{2}}

约分:

\frac{2 + 7}{2}

= a

化简

\frac{- 3}{2 ( - \frac{2 + 7}{2} )} : \frac{3}{2 2 + 7}

\frac{- 3}{2 ( - \frac{2 + 7}{2} )}

去除括号:

(- a) = - a

= \frac{- 3}{- 2 \frac{2 + 7}{2}}

使用分式法则:

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

= \frac{3}{2 \frac{2 + 7}{2}}

\frac{2 + 7}{2} = \frac{2 + 7}{2}

\frac{2 + 7}{2}

使用根式运算法则:

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

假定

a \geq 0, b \geq 0

= \frac{2 + 7}{2}

= \frac{3}{2 \cdot \frac{2 + 7}{2}}

乘

2 \cdot \frac{2 + 7}{2} : 2 2 + 7

2 \cdot \frac{2 + 7}{2}

分式相乘:

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

= \frac{2 + 7 \cdot 2}{2}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{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^{- \frac{1}{2} + 1} 2 + 7

数字相减:

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

= 2^{\frac{1}{2}} 2 + 7

使用根式运算法则:

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

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

= 2 2 + 7

= \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

a = \frac{3}{2 2 + 7}

将解代入原方程进行验证

将它们代入

a^{2} - b^{2} = - 2

检验解是否符合

去除与方程不符的解。

检验

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

的解:

真

a^{2} - b^{2} = - 2

代入

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

(\frac{3}{2 2 + 7})^{2} - - \frac{2 + 7}{2}^{2} = - 2

整理后得

- 2 = - 2

真

检验

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

的解:

真

a^{2} - b^{2} = - 2

代入

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

(- \frac{3}{2 2 + 7})^{2} - \frac{2 + 7}{2}^{2} = - 2

整理后得

- 2 = - 2

真

将它们代入

2 ab = - 3

检验解是否符合

去除与方程不符的解。

检验

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

的解:

真

2 ab = - 3

代入

a = \frac{3}{2 2 + 7}, b = - \frac{2 + 7}{2}

2 \cdot \frac{3}{2 2 + 7} - \frac{2 + 7}{2} = - 3

整理后得

- 3 = - 3

真

检验

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

的解:

真

2 ab = - 3

代入

a = - \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2}

2 (- \frac{3}{2 2 + 7}) \frac{2 + 7}{2} = - 3

整理后得

- 3 = - 3

真

因而，

a^{2} - b^{2} = - 2, 2 ab = - 3

最后的解是

a = - \frac{3}{2 2 + 7}, a = \frac{3}{2 2 + 7}, b = \frac{2 + 7}{2} b = - \frac{2 + 7}{2}

u = a + bi

代回

u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

解为

u = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, u = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, u = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

u = cos (x)

代回

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i, cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i, cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i :

无解

cos (x) = \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

化简

\frac{3}{2 2 + 7} + \frac{2 + 7}{2} i : \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3} + i \frac{2 + 7}{2}

\frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

乘以共轭根式

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

使用根式运算法则:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

使用根式运算法则:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

乘以共轭根式

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

使用根式运算法则:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

数字相乘:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

乘以共轭根式

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

使用平方差公式:

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

a = 4, b = 27

= 4^{2} - (27)^{2}

化简

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

使用指数法则:

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

= 2^{2} (7)^{2}

(7)^{2} : 7

使用根式运算法则:

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

= (7^{\frac{1}{2}})^{2}

使用指数法则:

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

= 7^{\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

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= 16 - 28

数字相减:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

使用分式法则:

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

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

消掉

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

分解

4 - 27 : 2 (2 - 7)

4 - 27

改写为

= 2 \cdot 2 - 27

因式分解出通项

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

约分:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6} + i \frac{2 + 7}{2}

将

- \frac{6 ( 2 - 7 ) 2 + 7}{6} + \frac{2 + 7}{2} i

改写成标准复数形式:

\frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

- \frac{6 ( 2 - 7 ) 2 + 7}{6} + \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

使用根式运算法则:

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

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

使用指数法则:

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

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

使用根式运算法则:

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

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

乘开

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

使用分配律:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

使用根式运算法则:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

乘开

7 (2 + 7) : 14 + 77

7 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

数字相乘:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= - \frac{2 2 + 7 - 14 + 7 7}{6} + i \frac{2 + 7}{2}

使用分式法则:

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

\frac{2 2 + 7 - 14 + 7 7}{6} = - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6})

= - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6}) + i \frac{2 + 7}{2}

去除括号:

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

= - \frac{2 2 + 7}{6} + \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

消掉

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

分解

6 : 23

因式分解

6 = 2 \cdot 3

= 2 \cdot 3

使用根式运算法则:

n ab = n a n b

= 23

= \frac{2 2 + 7}{2 3}

消掉

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

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}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

数字相减:

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

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

将复数的实部和虚部分组

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} + \frac{2 + 7}{2} i

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} = \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6}

将项转换为分式:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= - \frac{2 2 + 7}{3} + \frac{\frac{14 + 7 7}{6} 3}{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{- 2 2 + 7 + \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

使用根式运算法则:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

分式相乘:

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

= \frac{( 14 + 7 7 ) \cdot 3}{6}

约分:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

有理化:

\frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

乘以共轭根式

\frac{3}{3}

= \frac{( - 2 2 + 7 + \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

无解

cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i :

无解

cos (x) = - \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

化简

- \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i : \frac{3 ( - \frac{14 + 7 7}{2} + 2 2 + 7 )}{3} - i \frac{2 + 7}{2}

- \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

乘以共轭根式

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

使用根式运算法则:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

使用根式运算法则:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

乘以共轭根式

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

使用根式运算法则:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

数字相乘:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

乘以共轭根式

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

使用平方差公式:

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

a = 4, b = 27

= 4^{2} - (27)^{2}

化简

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

使用指数法则:

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

= 2^{2} (7)^{2}

(7)^{2} : 7

使用根式运算法则:

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

= (7^{\frac{1}{2}})^{2}

使用指数法则:

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

= 7^{\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

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= 16 - 28

数字相减:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

使用分式法则:

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

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

消掉

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

分解

4 - 27 : 2 (2 - 7)

4 - 27

改写为

= 2 \cdot 2 - 27

因式分解出通项

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

约分:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) - i \frac{2 + 7}{2}

将

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) - \frac{2 + 7}{2} i

改写成标准复数形式:

\frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) - \frac{2 + 7}{2} i

使用法则

- (- a) = a

= \frac{6 ( 2 - 7 ) 2 + 7}{6} - \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

使用根式运算法则:

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

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

使用指数法则:

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

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

使用根式运算法则:

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

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

乘开

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

使用分配律:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

使用根式运算法则:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

乘开

7 (2 + 7) : 14 + 77

7 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

数字相乘:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= \frac{2 2 + 7 - 14 + 7 7}{6} - i \frac{2 + 7}{2}

使用分式法则:

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

\frac{2 2 + 7 - 14 + 7 7}{6} = \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6}

= \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

消掉

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

分解

6 : 23

因式分解

6 = 2 \cdot 3

= 2 \cdot 3

使用根式运算法则:

n ab = n a n b

= 23

= \frac{2 2 + 7}{2 3}

消掉

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

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}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

数字相减:

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

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

将复数的实部和虚部分组

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} - \frac{2 + 7}{2} i

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} = \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6}

将项转换为分式:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= \frac{2 2 + 7}{3} - \frac{\frac{14 + 7 7}{6} 3}{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{2 2 + 7 - \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

使用根式运算法则:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

分式相乘:

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

= \frac{( 14 + 7 7 ) \cdot 3}{6}

约分:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

有理化:

\frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

乘以共轭根式

\frac{3}{3}

= \frac{( 2 2 + 7 - \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

无解

cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i :

无解

cos (x) = - \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

化简

- \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i : \frac{3 ( - \frac{14 + 7 7}{2} + 2 2 + 7 )}{3} + i \frac{2 + 7}{2}

- \frac{3}{2 2 + 7} + \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

乘以共轭根式

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

使用根式运算法则:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

使用根式运算法则:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

乘以共轭根式

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

使用根式运算法则:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

数字相乘:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

乘以共轭根式

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

使用平方差公式:

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

a = 4, b = 27

= 4^{2} - (27)^{2}

化简

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

使用指数法则:

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

= 2^{2} (7)^{2}

(7)^{2} : 7

使用根式运算法则:

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

= (7^{\frac{1}{2}})^{2}

使用指数法则:

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

= 7^{\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

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= 16 - 28

数字相减:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

使用分式法则:

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

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

消掉

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

分解

4 - 27 : 2 (2 - 7)

4 - 27

改写为

= 2 \cdot 2 - 27

因式分解出通项

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

约分:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) + i \frac{2 + 7}{2}

将

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) + \frac{2 + 7}{2} i

改写成标准复数形式:

\frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

- (- \frac{6 ( 2 - 7 ) 2 + 7}{6}) + \frac{2 + 7}{2} i

使用法则

- (- a) = a

= \frac{6 ( 2 - 7 ) 2 + 7}{6} + \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

使用根式运算法则:

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

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

使用指数法则:

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

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

使用根式运算法则:

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

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

乘开

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

使用分配律:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

使用根式运算法则:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

乘开

7 (2 + 7) : 14 + 77

7 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

数字相乘:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= \frac{2 2 + 7 - 14 + 7 7}{6} + i \frac{2 + 7}{2}

使用分式法则:

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

\frac{2 2 + 7 - 14 + 7 7}{6} = \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6}

= \frac{2 2 + 7}{6} - \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

消掉

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

分解

6 : 23

因式分解

6 = 2 \cdot 3

= 2 \cdot 3

使用根式运算法则:

n ab = n a n b

= 23

= \frac{2 2 + 7}{2 3}

消掉

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

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}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

数字相减:

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

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} + i \frac{2 + 7}{2}

将复数的实部和虚部分组

= \frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} + \frac{2 + 7}{2} i

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6} = \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7}{3} - \frac{14 + 7 7}{6}

将项转换为分式:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= \frac{2 2 + 7}{3} - \frac{\frac{14 + 7 7}{6} 3}{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{2 2 + 7 - \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

使用根式运算法则:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

分式相乘:

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

= \frac{( 14 + 7 7 ) \cdot 3}{6}

约分:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

有理化:

\frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

\frac{2 2 + 7 - \frac{14 + 7 7}{2}}{3}

乘以共轭根式

\frac{3}{3}

= \frac{( 2 2 + 7 - \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

= \frac{3 ( 2 2 + 7 - \frac{14 + 7 7}{2} )}{3} + \frac{2 + 7}{2} i

无解

cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i :

无解

cos (x) = \frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

化简

\frac{3}{2 2 + 7} - \frac{2 + 7}{2} i : \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3} - i \frac{2 + 7}{2}

\frac{3}{2 2 + 7} - \frac{2 + 7}{2} i

\frac{3}{2 2 + 7} = - \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{3}{2 2 + 7}

乘以共轭根式

\frac{2}{2}

= \frac{3 2}{2 2 + 7 2}

32 = 6

32

使用根式运算法则:

a b = a \cdot b

32 = 3 \cdot 2

= 3 \cdot 2

数字相乘:

3 \cdot 2 = 6

= 6

2 2 + 7 2 = 2 2 + 7

2 2 + 7 2

使用根式运算法则:

a a = a

22 = 2

= 2 2 + 7

= \frac{6}{2 2 + 7}

乘以共轭根式

\frac{2 + 7}{2 + 7}

= \frac{6 2 + 7}{2 2 + 7 2 + 7}

2 2 + 7 2 + 7 = 4 + 27

2 2 + 7 2 + 7

使用根式运算法则:

a a = a

2 + 7 2 + 7 = 2 + 7

= 2 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 2, b = 2, c = 7

= 2 \cdot 2 + 27

数字相乘:

2 \cdot 2 = 4

= 4 + 27

= \frac{6 2 + 7}{4 + 2 7}

乘以共轭根式

\frac{4 - 2 7}{4 - 2 7}

= \frac{6 2 + 7 ( 4 - 2 7 )}{( 4 + 2 7 ) ( 4 - 2 7 )}

(4 + 27) (4 - 27) = - 12

(4 + 27) (4 - 27)

使用平方差公式:

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

a = 4, b = 27

= 4^{2} - (27)^{2}

化简

4^{2} - (27)^{2} : - 12

4^{2} - (27)^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

(27)^{2} = 28

(27)^{2}

使用指数法则:

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

= 2^{2} (7)^{2}

(7)^{2} : 7

使用根式运算法则:

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

= (7^{\frac{1}{2}})^{2}

使用指数法则:

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

= 7^{\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

= 7

= 2^{2} \cdot 7

2^{2} = 4

= 4 \cdot 7

数字相乘:

4 \cdot 7 = 28

= 28

= 16 - 28

数字相减:

16 - 28 = - 12

= - 12

= - 12

= \frac{6 ( 4 - 2 7 ) 2 + 7}{- 12}

使用分式法则:

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

= - \frac{6 ( 4 - 2 7 ) 2 + 7}{12}

消掉

\frac{6 ( 4 - 2 7 ) 2 + 7}{12} : \frac{6 ( 2 - 7 ) 2 + 7}{6}

\frac{6 ( 4 - 2 7 ) 2 + 7}{12}

分解

4 - 27 : 2 (2 - 7)

4 - 27

改写为

= 2 \cdot 2 - 27

因式分解出通项

2

= 2 (2 - 7)

= \frac{6 \cdot 2 ( 2 - 7 ) 2 + 7}{12}

约分:

2

= \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6}

= - \frac{6 ( 2 - 7 ) 2 + 7}{6} - i \frac{2 + 7}{2}

将

- \frac{6 ( 2 - 7 ) 2 + 7}{6} - \frac{2 + 7}{2} i

改写成标准复数形式:

\frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

- \frac{6 ( 2 - 7 ) 2 + 7}{6} - \frac{2 + 7}{2} i

\frac{6 ( 2 - 7 ) 2 + 7}{6} = \frac{2 2 + 7 - 14 + 7 7}{6}

\frac{6 ( 2 - 7 ) 2 + 7}{6}

使用根式运算法则:

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

6 = 6^{\frac{1}{2}}

= \frac{6 ^{\frac{1}{2}} ( 2 - 7 ) 2 + 7}{6}

使用指数法则:

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

\frac{6 ^{\frac{1}{2}}}{6 ^{1}} = \frac{1}{6 ^{1 - \frac{1}{2}}}

= \frac{( 2 - 7 ) 2 + 7}{6 ^{1 - \frac{1}{2}}}

数字相减:

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

= \frac{( 2 - 7 ) 2 + 7}{6 ^{\frac{1}{2}}}

使用根式运算法则:

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

6^{\frac{1}{2}} = 6

= \frac{( 2 - 7 ) 2 + 7}{6}

乘开

(2 - 7) 2 + 7 : 2 2 + 7 - 14 + 77

(2 - 7) 2 + 7

使用分配律:

a (b - c) = ab - a c

a = 2 + 7, b = 2, c = 7

= 2 + 7 \cdot 2 - 2 + 7 7

= 2 2 + 7 - 7 2 + 7

7 2 + 7 = 14 + 77

7 2 + 7

使用根式运算法则:

a b = a \cdot b

7 2 + 7 = 7 (2 + 7)

= 7 (2 + 7)

乘开

7 (2 + 7) : 14 + 77

7 (2 + 7)

使用分配律:

a (b + c) = ab + a c

a = 7, b = 2, c = 7

= 7 \cdot 2 + 77

数字相乘:

7 \cdot 2 = 14

= 14 + 77

= 14 + 77

= 2 2 + 7 - 14 + 77

= \frac{2 2 + 7 - 14 + 7 7}{6}

= - \frac{2 2 + 7 - 14 + 7 7}{6} - i \frac{2 + 7}{2}

使用分式法则:

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

\frac{2 2 + 7 - 14 + 7 7}{6} = - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6})

= - (\frac{2 2 + 7}{6}) - (- \frac{14 + 7 7}{6}) - i \frac{2 + 7}{2}

去除括号:

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

= - \frac{2 2 + 7}{6} + \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

消掉

\frac{2 2 + 7}{6} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{6}

分解

6 : 23

因式分解

6 = 2 \cdot 3

= 2 \cdot 3

使用根式运算法则:

n ab = n a n b

= 23

= \frac{2 2 + 7}{2 3}

消掉

\frac{2 2 + 7}{2 3} : \frac{2 2 + 7}{3}

\frac{2 2 + 7}{2 3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{2 ^{\frac{1}{2}} 3}

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}}

= \frac{2 ^{- \frac{1}{2} + 1} 2 + 7}{3}

数字相减:

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

= \frac{2 ^{\frac{1}{2}} 2 + 7}{3}

使用根式运算法则:

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

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

= \frac{2 2 + 7}{3}

= \frac{2 2 + 7}{3}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

合并相同指数项 :

\frac{x}{y} = \frac{x}{y}

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} - i \frac{2 + 7}{2}

将复数的实部和虚部分组

= - \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} - \frac{2 + 7}{2} i

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6} = \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

- \frac{2 2 + 7}{3} + \frac{14 + 7 7}{6}

将项转换为分式:

\frac{14 + 7 7}{6} = \frac{\frac{14 + 7 \cdot 7}{6} 3}{3}

= - \frac{2 2 + 7}{3} + \frac{\frac{14 + 7 7}{6} 3}{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{- 2 2 + 7 + \frac{14 + 7 7}{6} 3}{3}

\frac{14 + 7 7}{6} 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} 3

使用根式运算法则:

a b = a \cdot b

3 \frac{14 + 7 7}{6} = 3 \cdot \frac{14 + 7 7}{6}

= 3 \cdot \frac{14 + 7 7}{6}

\frac{14 + 7 7}{6} \cdot 3 = \frac{14 + 7 7}{2}

\frac{14 + 7 7}{6} \cdot 3

分式相乘:

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

= \frac{( 14 + 7 7 ) \cdot 3}{6}

约分:

3

= \frac{14 + 7 7}{2}

= \frac{14 + 7 7}{2}

= \frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

有理化:

\frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

\frac{- 2 2 + 7 + \frac{14 + 7 7}{2}}{3}

乘以共轭根式

\frac{3}{3}

= \frac{( - 2 2 + 7 + \frac{14 + 7 7}{2} ) 3}{3 3}

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3}

= \frac{3 ( \frac{14 + 7 7}{2} - 2 2 + 7 )}{3}

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

= \frac{3 ( - 2 2 + 7 + \frac{14 + 7 7}{2} )}{3} - \frac{2 + 7}{2} i

无解

合并所有解

x \in R 无解

作图

流行的例子

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((cos^3(a)))/((2cos^2(a)-1))=cos(a)

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cos (x - 4 5^{\circ}) = 0

7 tan^{2} (x) - 15 = 0

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