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cot(52.5)

解答

cot (52. 5^{\circ})

解答

\frac{15 + 6 6 - 8 3 - 10 2}{15 + 6 6 - 8 3 - 10 2}

+1

十进制表示法

0.76732 \dots

求解步骤

cot (52. 5^{\circ})

使用三角恒等式改写:

\frac{1}{tan ( 52. 5 ^{\circ} )}

cot (52. 5^{\circ})

使用基本三角恒等式:

cot (x) = \frac{1}{tan ( x )}

= \frac{1}{tan ( 52. 5 ^{\circ} )}

= \frac{1}{tan ( 52. 5 ^{\circ} )}

使用三角恒等式改写:

tan (52. 5^{\circ}) = 15 + 66 - 83 - 102

tan (52. 5^{\circ})

使用三角恒等式改写:

\frac{1 - cos ( 10 5 ^{\circ} )}{1 + cos ( 10 5 ^{\circ} )}

tan (52. 5^{\circ})

将

tan (52. 5^{\circ})

写为

tan (\frac{10 5 ^{\circ}}{2})

= tan (\frac{10 5 ^{\circ}}{2})

使用半角公式:

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

使用三角恒等式改写:

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

利用以下特性

tan (θ) = \frac{sin ( θ )}{cos ( θ )}

两边进行平方

tan^{2} (θ) = \frac{sin ^{2} ( θ )}{cos ^{2} ( θ )}

使用三角恒等式改写:

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

使用倍角公式

cos (2 θ) = 1 - 2 sin^{2} (θ)

交换两边

2 sin^{2} (θ) - 1 = - cos (2 θ)

两边加上

1

2 sin^{2} (θ) = 1 - cos (2 θ)

两边除以

2

sin^{2} (θ) = \frac{1 - cos ( 2 θ )}{2}

使用三角恒等式改写:

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

使用倍角公式

cos (2 θ) = 2 cos^{2} (θ) - 1

交换两边

2 cos^{2} (θ) - 1 = cos (2 θ)

两边加上

1

2 sin^{2} (θ) = 1 + cos (2 θ)

两边除以

2

cos^{2} (θ) = \frac{1 + cos ( 2 θ )}{2}

tan^{2} (θ) = \frac{\frac{1 - c o s ( 2 θ )}{2}}{\frac{1 + c o s ( 2 θ )}{2}}

化简

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

用

\frac{θ}{2}

替代

θ

tan^{2} (\frac{θ}{2}) = \frac{1 - cos ( 2 \cdot \frac{θ}{2} )}{1 + cos ( 2 \cdot \frac{θ}{2} )}

化简

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

Square root both sides

Choose the root sign according to the quadrant of

\frac{θ}{2}

:

range [0, 9 0^{\circ}] [9 0^{\circ}, 18 0^{\circ}] quadrant I II tan positive negative

tan (\frac{θ}{2}) = \frac{1 - cos ( θ )}{1 + cos ( θ )}

= \frac{1 - cos ( 10 5 ^{\circ} )}{1 + cos ( 10 5 ^{\circ} )}

= \frac{1 - cos ( 10 5 ^{\circ} )}{1 + cos ( 10 5 ^{\circ} )}

使用三角恒等式改写:

cos (10 5^{\circ}) = \frac{2 ( 1 - 3 )}{4}

cos (10 5^{\circ})

使用三角恒等式改写:

cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

cos (10 5^{\circ})

将

cos (10 5^{\circ})

写为

cos (6 0^{\circ} + 4 5^{\circ})

= cos (6 0^{\circ} + 4 5^{\circ})

使用角和恒等式:

cos (s + t) = cos (s) cos (t) - sin (s) sin (t)

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

= cos (6 0^{\circ}) cos (4 5^{\circ}) - sin (6 0^{\circ}) sin (4 5^{\circ})

使用以下普通恒等式:

cos (6 0^{\circ}) = \frac{1}{2}

cos (6 0^{\circ})

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{1}{2}

使用以下普通恒等式:

cos (4 5^{\circ}) = \frac{2}{2}

cos (4 5^{\circ})

cos (x)

周期表（周期为

36 0^{\circ} n

）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} cos (x) 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} cos (x) - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2} 0 \frac{1}{2} \frac{2}{2} \frac{3}{2}

= \frac{2}{2}

使用以下普通恒等式:

sin (6 0^{\circ}) = \frac{3}{2}

sin (6 0^{\circ})

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{3}{2}

使用以下普通恒等式:

sin (4 5^{\circ}) = \frac{2}{2}

sin (4 5^{\circ})

sin (x)

周期表（周期为

36 0^{\circ} n

"）：

x 0 3 0^{\circ} 4 5^{\circ} 6 0^{\circ} 9 0^{\circ} 12 0^{\circ} 13 5^{\circ} 15 0^{\circ} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x 18 0^{\circ} 21 0^{\circ} 22 5^{\circ} 24 0^{\circ} 27 0^{\circ} 30 0^{\circ} 31 5^{\circ} 33 0^{\circ} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= \frac{2}{2}

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

化简

\frac{1}{2} \cdot \frac{2}{2} - \frac{3}{2} \cdot \frac{2}{2} : \frac{2 ( 1 - 3 )}{4}

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

因式分解出通项

\frac{2}{2}

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

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

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

使用法则

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

= \frac{1 - 3}{2}

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

分式相乘:

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

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

数字相乘:

2 \cdot 2 = 4

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

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

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

化简

\frac{1 - \frac{2 ( 1 - 3 )}{4}}{1 + \frac{2 ( 1 - 3 )}{4}} : 15 + 66 - 83 - 102

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

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

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

化简

1 + \frac{2 ( 1 - 3 )}{4} : \frac{4 + 2 ( 1 - 3 )}{4}

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

将项转换为分式:

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

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

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 4 + 2 ( 1 - 3 )}{4}

数字相乘:

1 \cdot 4 = 4

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

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

化简

1 - \frac{2 ( 1 - 3 )}{4} : \frac{4 - 2 ( 1 - 3 )}{4}

1 - \frac{2 ( 1 - 3 )}{4}

将项转换为分式:

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

= \frac{1 \cdot 4}{4} - \frac{2 ( 1 - 3 )}{4}

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 4 - 2 ( 1 - 3 )}{4}

数字相乘:

1 \cdot 4 = 4

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

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

分式相除:

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

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

约分:

4

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

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

\frac{4 - 2 ( 1 - 3 )}{4 + 2 ( 1 - 3 )} = 15 + 66 - 83 - 102

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

乘以共轭根式

\frac{4 - 2 ( 1 - 3 )}{4 - 2 ( 1 - 3 )}

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

(4 - 2 (1 - 3)) (4 - 2 (1 - 3)) = 24 + 86 - 82 - 43

(4 - 2 (1 - 3)) (4 - 2 (1 - 3))

使用指数法则:

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

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

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

数字相加:

1 + 1 = 2

= (4 - 2 (1 - 3))^{2}

使用完全平方公式:

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

a = 4, b = 2 (1 - 3)

= 4^{2} - 2 \cdot 42 (1 - 3) + (2 (1 - 3))^{2}

化简

4^{2} - 2 \cdot 42 (1 - 3) + (2 (1 - 3))^{2} : 24 + 86 - 82 - 43

4^{2} - 2 \cdot 42 (1 - 3) + (2 (1 - 3))^{2}

4^{2} = 16

4^{2}

4^{2} = 16

= 16

2 \cdot 42 (1 - 3) = 82 (1 - 3)

2 \cdot 42 (1 - 3)

数字相乘:

2 \cdot 4 = 8

= 82 (1 - 3)

(2 (1 - 3))^{2} = 2 (4 - 23)

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

使用指数法则:

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

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

= 2 (1 - 3)^{2}

(1 - 3)^{2} = 4 - 23

(1 - 3)^{2}

使用完全平方公式:

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

a = 1, b = 3

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

化简

1^{2} - 2 \cdot 1 \cdot 3 + (3)^{2} : 4 - 23

1^{2} - 2 \cdot 1 \cdot 3 + (3)^{2}

使用法则

1^{a} = 1

1^{2} = 1

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

2 \cdot 1 \cdot 3 = 23

2 \cdot 1 \cdot 3

数字相乘:

2 \cdot 1 = 2

= 23

(3)^{2} = 3

(3)^{2}

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

= 1 - 23 + 3

数字相加:

1 + 3 = 4

= 4 - 23

= 4 - 23

= 2 (4 - 23)

= 16 - 82 (1 - 3) + 2 (4 - 23)

乘开

- 82 (1 - 3) : - 82 + 86

- 82 (1 - 3)

使用分配律:

a (b - c) = ab - a c

a = - 82, b = 1, c = 3

= - 82 \cdot 1 - (- 82) 3

使用加减运算法则

- (- a) = a

= - 8 \cdot 1 \cdot 2 + 823

化简

- 8 \cdot 1 \cdot 2 + 823 : - 82 + 86

- 8 \cdot 1 \cdot 2 + 823

8 \cdot 1 \cdot 2 = 82

8 \cdot 1 \cdot 2

数字相乘:

8 \cdot 1 = 8

= 82

823 = 86

823

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 8 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 86

= - 82 + 86

= - 82 + 86

= 16 - 82 + 86 + 2 (4 - 23)

乘开

2 (4 - 23) : 8 - 43

2 (4 - 23)

使用分配律:

a (b - c) = ab - a c

a = 2, b = 4, c = 23

= 2 \cdot 4 - 2 \cdot 23

化简

2 \cdot 4 - 2 \cdot 23 : 8 - 43

2 \cdot 4 - 2 \cdot 23

数字相乘:

2 \cdot 4 = 8

= 8 - 2 \cdot 23

数字相乘:

2 \cdot 2 = 4

= 8 - 43

= 8 - 43

= 16 - 82 + 86 + 8 - 43

数字相加:

16 + 8 = 24

= 24 + 86 - 82 - 43

= 24 + 86 - 82 - 43

(4 + 2 (1 - 3)) (4 - 2 (1 - 3)) = 8 + 43

(4 + 2 (1 - 3)) (4 - 2 (1 - 3))

乘开

4 + 2 (1 - 3) : 4 + 2 - 6

4 + 2 (1 - 3)

乘开

2 (1 - 3) : 2 - 6

2 (1 - 3)

使用分配律:

a (b - c) = ab - a c

a = 2, b = 1, c = 3

= 2 \cdot 1 - 23

= 1 \cdot 2 - 23

化简

1 \cdot 2 - 23 : 2 - 6

1 \cdot 2 - 23

1 \cdot 2 = 2

1 \cdot 2

乘以:

1 \cdot 2 = 2

= 2

23 = 6

23

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

= 2 - 6

= 2 - 6

= 4 + 2 - 6

= (4 + 2 - 6) (- 2 (1 - 3) + 4)

乘开

4 - 2 (1 - 3) : 4 - 2 + 6

4 - 2 (1 - 3)

乘开

- 2 (1 - 3) : - 2 + 6

- 2 (1 - 3)

使用分配律:

a (b - c) = ab - a c

a = - 2, b = 1, c = 3

= - 2 \cdot 1 - (- 2) 3

使用加减运算法则

- (- a) = a

= - 1 \cdot 2 + 23

化简

- 1 \cdot 2 + 23 : - 2 + 6

- 1 \cdot 2 + 23

1 \cdot 2 = 2

1 \cdot 2

乘以:

1 \cdot 2 = 2

= 2

23 = 6

23

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 6

= - 2 + 6

= - 2 + 6

= 4 - 2 + 6

= (4 + 2 - 6) (4 + 6 - 2)

打开括号

= 4 \cdot 4 + 4 (- 2) + 46 + 2 \cdot 4 + 2 (- 2) + 26 + (- 6) \cdot 4 + (- 6) (- 2) + (- 6) 6

使用加减运算法则

+ (- a) = - a, (- a) (- b) = ab

= 4 \cdot 4 - 42 + 46 + 42 - 22 + 26 - 46 + 62 - 66

化简

4 \cdot 4 - 42 + 46 + 42 - 22 + 26 - 46 + 62 - 66 : 8 + 43

4 \cdot 4 - 42 + 46 + 42 - 22 + 26 - 46 + 62 - 66

同类项相加:

26 + 62 = 262

= 4 \cdot 4 - 42 + 46 + 42 - 22 + 262 - 46 - 66

同类项相加:

- 42 + 42 = 0

= 4 \cdot 4 + 46 - 22 + 262 - 46 - 66

同类项相加:

46 - 46 = 0

= 4 \cdot 4 - 22 + 262 - 66

4 \cdot 4 = 16

4 \cdot 4

数字相乘:

4 \cdot 4 = 16

= 16

22 = 2

22

使用根式运算法则:

a a = a

22 = 2

= 2

262 = 43

262

使用指数法则:

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

22 = 2 \cdot 2^{\frac{1}{2}} = 2^{1 + \frac{1}{2}}

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

分解整数

6 = 2 \cdot 3

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

使用根式运算法则:

2 \cdot 3 = 23

= 23 \cdot 2^{1 + \frac{1}{2}}

使用指数法则:

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

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

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

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

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

合并分式

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

使用法则

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

= \frac{1 + 1}{2}

数字相加:

1 + 1 = 2

= \frac{2}{2}

使用法则

\frac{a}{a} = 1

= 1

= 2^{1 + 1}

数字相加:

1 + 1 = 2

= 2^{2}

= 2^{2} 3

2^{2} = 4

= 43

66 = 6

66

使用根式运算法则:

a a = a

66 = 6

= 6

= 16 - 2 + 43 - 6

数字相减:

16 - 2 - 6 = 8

= 8 + 43

= 8 + 43

= \frac{24 + 8 6 - 8 2 - 4 3}{8 + 4 3}

分解

24 + 86 - 82 - 43 : 4 (6 + 26 - 22 - 3)

24 + 86 - 82 - 43

改写为

= 4 \cdot 6 + 4 \cdot 26 - 4 \cdot 22 - 43

因式分解出通项

4

= 4 (6 + 26 - 22 - 3)

= \frac{4 ( 6 + 2 6 - 2 2 - 3 )}{8 + 4 3}

分解

8 + 43 : 4 (2 + 3)

8 + 43

改写为

= 4 \cdot 2 + 43

因式分解出通项

4

= 4 (2 + 3)

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

数字相除:

\frac{4}{4} = 1

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

去除括号:

(a) = a

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

乘以共轭根式

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

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

(6 + 26 - 22 - 3) (2 - 3) = 15 + 66 - 83 - 102

(6 + 26 - 22 - 3) (2 - 3)

打开括号

= 6 \cdot 2 + 6 (- 3) + 26 \cdot 2 + 26 (- 3) + (- 22) \cdot 2 + (- 22) (- 3) + (- 3) \cdot 2 + (- 3) (- 3)

使用加减运算法则

+ (- a) = - a, (- a) (- b) = ab

= 6 \cdot 2 - 63 + 2 \cdot 26 - 263 - 2 \cdot 22 + 223 - 23 + 33

化简

6 \cdot 2 - 63 + 2 \cdot 26 - 263 - 2 \cdot 22 + 223 - 23 + 33 : 15 + 66 - 83 - 102

6 \cdot 2 - 63 + 2 \cdot 26 - 263 - 2 \cdot 22 + 223 - 23 + 33

同类项相加:

- 63 - 23 = - 83

= 6 \cdot 2 - 83 + 2 \cdot 26 - 263 - 2 \cdot 22 + 223 + 33

6 \cdot 2 = 12

6 \cdot 2

数字相乘:

6 \cdot 2 = 12

= 12

2 \cdot 26 = 46

2 \cdot 26

数字相乘:

2 \cdot 2 = 4

= 46

263 = 62

263

分解整数

6 = 3 \cdot 2

= 2 3 \cdot 2 3

使用根式运算法则:

3 \cdot 2 = 32

= 2323

使用根式运算法则:

a a = a

33 = 3

= 2 \cdot 32

数字相乘:

2 \cdot 3 = 6

= 62

2 \cdot 22 = 42

2 \cdot 22

数字相乘:

2 \cdot 2 = 4

= 42

223 = 26

223

使用根式运算法则:

a b = a \cdot b

23 = 2 \cdot 3

= 2 2 \cdot 3

数字相乘:

2 \cdot 3 = 6

= 26

33 = 3

33

使用根式运算法则:

a a = a

33 = 3

= 3

= 12 - 83 + 46 - 62 - 42 + 26 + 3

同类项相加:

- 62 - 42 = - 102

= 12 - 83 + 46 - 102 + 26 + 3

同类项相加:

46 + 26 = 66

= 12 - 83 + 66 - 102 + 3

数字相加:

12 + 3 = 15

= 15 + 66 - 83 - 102

= 15 + 66 - 83 - 102

(2 + 3) (2 - 3) = 1

(2 + 3) (2 - 3)

使用平方差公式:

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

a = 2, b = 3

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

化简

2^{2} - (3)^{2} : 1

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

2^{2} = 4

2^{2}

2^{2} = 4

= 4

(3)^{2} = 3

(3)^{2}

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

= 4 - 3

数字相减:

4 - 3 = 1

= 1

= 1

= \frac{15 + 6 6 - 8 3 - 10 2}{1}

使用法则

\frac{a}{1} = a

= 15 + 66 - 83 - 102

= 15 + 66 - 83 - 102

= 15 + 66 - 83 - 102

= \frac{1}{15 + 6 6 - 8 3 - 10 2}

化简

\frac{1}{15 + 6 6 - 8 3 - 10 2} : \frac{15 + 6 6 - 8 3 - 10 2}{15 + 6 6 - 8 3 - 10 2}

\frac{1}{15 + 6 6 - 8 3 - 10 2}

乘以共轭根式

\frac{15 + 6 6 - 8 3 - 10 2}{15 + 6 6 - 8 3 - 10 2}

= \frac{1 \cdot 15 + 6 6 - 8 3 - 10 2}{15 + 6 6 - 8 3 - 10 2 15 + 6 6 - 8 3 - 10 2}

1 \cdot 15 + 66 - 83 - 102 = 15 + 66 - 83 - 102

15 + 66 - 83 - 102 15 + 66 - 83 - 102 = 15 + 6^{\frac{3}{2}} - 83 - 102

15 + 66 - 83 - 102 15 + 66 - 83 - 102

使用根式运算法则:

a a = a

15 + 66 - 83 - 102 15 + 66 - 83 - 102 = 15 + 66 - 83 - 102

= 15 + 66 - 83 - 102

66 = 6^{\frac{3}{2}}

66

使用指数法则:

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

66 = 6 \cdot 6^{\frac{1}{2}} = 6^{1 + \frac{1}{2}}

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

化简

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

1 + \frac{1}{2}

将项转换为分式:

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

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

1 \cdot 2 + 1 = 3

1 \cdot 2 + 1

数字相乘:

1 \cdot 2 = 2

= 2 + 1

数字相加:

2 + 1 = 3

= 3

= \frac{3}{2}

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

= 15 + 6^{\frac{3}{2}} - 83 - 102

= \frac{15 + 6 6 - 8 3 - 10 2}{15 + 6 ^{\frac{3}{2}} - 8 3 - 10 2}

6^{\frac{3}{2}} = 66

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

6^{\frac{3}{2}} = 6^{1 + \frac{1}{2}}

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

使用指数法则:

x^{a + b} = x^{a} x^{b}

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

整理后得

= 66

= \frac{15 + 6 6 - 8 3 - 10 2}{15 + 6 6 - 8 3 - 10 2}

= \frac{15 + 6 6 - 8 3 - 10 2}{15 + 6 6 - 8 3 - 10 2}

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