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人気のある三角関数 >

180tan(72)+240

解

180 tan (7 2^{\circ}) + 240

解

1352 5 - 5 + 4510 5 - 5 + 240

+1

十進法表記

793.98303 \dots

解答ステップ

180 tan (7 2^{\circ}) + 240

三角関数の公式を使用して書き換える:

tan (7 2^{\circ}) = \frac{2 tan ( 3 6 ^{\circ} )}{1 - tan ^{2} ( 3 6 ^{\circ} )}

tan (7 2^{\circ})

tan (7 2^{\circ})

を以下として書く:

tan (2 \cdot 3 6^{\circ})

= tan (2 \cdot 3 6^{\circ})

2倍角の公式を使用:

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

= \frac{2 tan ( 3 6 ^{\circ} )}{1 - tan ^{2} ( 3 6 ^{\circ} )}

= 180 \cdot \frac{2 tan ( 3 6 ^{\circ} )}{1 - tan ^{2} ( 3 6 ^{\circ} )} + 240

簡素化

= \frac{360 tan ( 3 6 ^{\circ} ) + 240 ( 1 - tan ^{2} ( 3 6 ^{\circ} ) )}{1 - tan ^{2} ( 3 6 ^{\circ} )}

三角関数の公式を使用して書き換える:

tan (3 6^{\circ}) = \frac{2 ( 5 - 1 ) 5 - 5}{4}

tan (3 6^{\circ})

三角関数の公式を使用して書き換える:

\frac{sin ( 3 6 ^{\circ} )}{cos ( 3 6 ^{\circ} )}

tan (3 6^{\circ})

基本的な三角関数の公式を使用する:

tan (x) = \frac{sin ( x )}{cos ( x )}

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

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

三角関数の公式を使用して書き換える:

sin (3 6^{\circ}) = \frac{2 5 - 5}{4}

sin (3 6^{\circ})

以下を証明する:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

加法定理に次の積を使用する:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

以下を証明する:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

2倍角の公式を使用:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

次の恒等を使用する:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

以下で両辺を割る

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代用

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

以下を証明する:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

因数分解の規則を使用する:

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

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

改良

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

以下を証明する:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

2倍角の公式を使用:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

次の恒等を使用する:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

以下で両辺を割る

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代用

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

代用

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

改良

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

両辺に

\frac{1}{4}

を足す

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

改良

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

用側の平方根を取得する

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

負の数にはできない

sin (1 8^{\circ})

負の数にはできない

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

次のequationを追加する

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

改良

cos (3 6^{\circ}) = \frac{5 + 1}{4}

両辺を2乗する

(cos (3 6^{\circ}))^{2} = (\frac{5 + 1}{4})^{2}

次の恒等を使用する:

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

sin^{2} (3 6^{\circ}) = 1 - cos^{2} (3 6^{\circ})

代用

cos (3 6^{\circ}) = \frac{5 + 1}{4}

sin^{2} (3 6^{\circ}) = 1 - (\frac{5 + 1}{4})^{2}

改良

sin^{2} (3 6^{\circ}) = \frac{5 - 5}{8}

用側の平方根を取得する

sin (3 6^{\circ}) = \pm \frac{5 - 5}{8}

sin (3 6^{\circ})

負の数にはできない

sin (3 6^{\circ}) = \frac{5 - 5}{8}

改良

sin (3 6^{\circ}) = \frac{\frac{5 - 5}{2}}{2}

= \frac{\frac{5 - 5}{2}}{2}

簡素化

= \frac{2 5 - 5}{4}

三角関数の公式を使用して書き換える:

cos (3 6^{\circ}) = \frac{5 + 1}{4}

cos (3 6^{\circ})

以下を証明する:

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

加法定理に次の積を使用する:

2 sin (x) cos (y) = sin (x + y) - sin (x - y)

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = sin (5 4^{\circ}) - sin (1 8^{\circ})

以下を証明する:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

2倍角の公式を使用:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

次の恒等を使用する:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

以下で両辺を割る

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代用

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

\frac{1}{2} = sin (5 4^{\circ}) - sin (1 8^{\circ})

sin (5 4^{\circ}) = cos (9 0^{\circ} - 5 4^{\circ})

\frac{1}{2} = cos (9 0^{\circ} - 5 4^{\circ}) - sin (1 8^{\circ})

\frac{1}{2} = cos (3 6^{\circ}) - sin (1 8^{\circ})

以下を証明する:

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

因数分解の規則を使用する:

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

a = cos (3 6^{\circ}) + sin (1 8^{\circ})

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = ((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) ((cos (3 6^{\circ}) + sin (1 8^{\circ})) - (cos (3 6^{\circ}) - sin (1 8^{\circ})))

改良

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 2 (2 cos (3 6^{\circ}) sin (1 8^{\circ}))

以下を証明する:

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

2倍角の公式を使用:

sin (2 x) = 2 sin (x) cos (x)

sin (7 2^{\circ}) = 2 sin (3 6^{\circ}) cos (3 6^{\circ})

sin (7 2^{\circ}) sin (3 6^{\circ}) = 4 sin (3 6^{\circ}) sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

sin (3 6^{\circ})

sin (7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

次の恒等を使用する:

sin (x) = cos (9 0^{\circ} - x)

sin (7 2^{\circ}) = cos (9 0^{\circ} - 7 2^{\circ})

cos (9 0^{\circ} - 7 2^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

cos (1 8^{\circ}) = 4 sin (1 8^{\circ}) cos (3 6^{\circ}) cos (1 8^{\circ})

以下で両辺を割る

cos (1 8^{\circ})

1 = 4 sin (1 8^{\circ}) cos (3 6^{\circ})

以下で両辺を割る

2

\frac{1}{2} = 2 sin (1 8^{\circ}) cos (3 6^{\circ})

代用

2 cos (3 6^{\circ}) sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (cos (3 6^{\circ}) - sin (1 8^{\circ}))^{2} = 1

代用

cos (3 6^{\circ}) - sin (1 8^{\circ}) = \frac{1}{2}

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - (\frac{1}{2})^{2} = 1

改良

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} = 1

両辺に

\frac{1}{4}

を足す

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} - \frac{1}{4} + \frac{1}{4} = 1 + \frac{1}{4}

改良

(cos (3 6^{\circ}) + sin (1 8^{\circ}))^{2} = \frac{5}{4}

用側の平方根を取得する

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \pm \frac{5}{4}

cos (3 6^{\circ})

負の数にはできない

sin (1 8^{\circ})

負の数にはできない

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{4}

次のequationを追加する

cos (3 6^{\circ}) + sin (1 8^{\circ}) = \frac{5}{2}

((cos (3 6^{\circ}) + sin (1 8^{\circ})) + (cos (3 6^{\circ}) - sin (1 8^{\circ}))) = (\frac{5}{2} + \frac{1}{2})

改良

cos (3 6^{\circ}) = \frac{5 + 1}{4}

= \frac{5 + 1}{4}

= \frac{\frac{2 5 - 5}{4}}{\frac{5 + 1}{4}}

簡素化

\frac{\frac{2 5 - 5}{4}}{\frac{5 + 1}{4}} : \frac{2 ( 5 - 1 ) 5 - 5}{4}

\frac{\frac{2 5 - 5}{4}}{\frac{5 + 1}{4}}

分数を割る:

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

= \frac{2 5 - 5 \cdot 4}{4 ( 5 + 1 )}

共通因数を約分する:

4

= \frac{2 5 - 5}{5 + 1}

有理化する

\frac{2 5 - 5}{5 + 1} : \frac{2 ( 5 - 1 ) 5 - 5}{4}

\frac{2 5 - 5}{5 + 1}

共役で乗じる

\frac{5 - 1}{5 - 1}

= \frac{2 5 - 5 ( 5 - 1 )}{( 5 + 1 ) ( 5 - 1 )}

(5 + 1) (5 - 1) = 4

(5 + 1) (5 - 1)

2乗の差の公式を適用する:

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

a = 5, b = 1

= (5)^{2} - 1^{2}

簡素化

(5)^{2} - 1^{2} : 4

(5)^{2} - 1^{2}

規則を適用

1^{a} = 1

1^{2} = 1

= (5)^{2} - 1

(5)^{2} = 5

(5)^{2}

累乗根の規則を適用する:

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

= (5^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= 5^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 5

= 5 - 1

数を引く:

5 - 1 = 4

= 4

= 4

= \frac{2 ( 5 - 1 ) 5 - 5}{4}

= \frac{2 ( 5 - 1 ) 5 - 5}{4}

= \frac{2 ( 5 - 1 ) 5 - 5}{4}

= \frac{360 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4} + 240 ( 1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} )}{1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2}}

簡素化

\frac{360 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4} + 240 ( 1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} )}{1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2}} : 1352 5 - 5 + 4510 5 - 5 + 240

\frac{360 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4} + 240 ( 1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} )}{1 - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2}}

(\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2} = 5 - 25

(\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2}

指数の規則を適用する:

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

= \frac{( 2 ( 5 - 1 ) 5 - 5 ) ^{2}}{4 ^{2}}

指数の規則を適用する:

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

(2 (5 - 1) 5 - 5)^{2} = (2)^{2} (5 - 5)^{2} (5 - 1)^{2}

= \frac{( 2 ) ^{2} ( 5 - 5 ) ^{2} ( 5 - 1 ) ^{2}}{4 ^{2}}

(2)^{2} : 2

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

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

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 2

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 ) ^{2}}{4 ^{2}}

(5 - 5)^{2} : 5 - 5

累乗根の規則を適用する:

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

= ((5 - 5)^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= (5 - 5)^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 5 - 5

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 )}{4 ^{2}}

因数

4^{2} : 2^{4}

因数

4 = 2^{2}

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

簡素化

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

(2^{2})^{2}

指数の規則を適用する:

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

= 2^{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= 2^{4}

= 2^{4}

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 )}{2 ^{4}}

共通因数を約分する:

2

= \frac{( 5 - 1 ) ^{2} ( 5 - 5 )}{2 ^{3}}

(5 - 1)^{2} = 6 - 25

(5 - 1)^{2}

完全平方式を適用する:

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

a = 5, b = 1

= (5)^{2} - 25 \cdot 1 + 1^{2}

簡素化

(5)^{2} - 25 \cdot 1 + 1^{2} : 6 - 25

(5)^{2} - 25 \cdot 1 + 1^{2}

規則を適用

1^{a} = 1

1^{2} = 1

= (5)^{2} - 2 \cdot 1 \cdot 5 + 1

(5)^{2} = 5

(5)^{2}

累乗根の規則を適用する:

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

= (5^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= 5^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 5

25 \cdot 1 = 25

25 \cdot 1

数を乗じる:

2 \cdot 1 = 2

= 25

= 5 - 25 + 1

数を足す:

5 + 1 = 6

= 6 - 25

= 6 - 25

= \frac{( 6 - 2 5 ) ( 5 - 5 )}{2 ^{3}}

因数

6 - 25 : 2 (3 - 5)

6 - 25

書き換え

= 2 \cdot 3 - 25

共通項をくくり出す

2

= 2 (3 - 5)

= \frac{2 ( 3 - 5 ) ( 5 - 5 )}{2 ^{3}}

共通因数を約分する:

2

= \frac{( 3 - 5 ) ( 5 - 5 )}{2 ^{2}}

拡張

(3 - 5) (5 - 5) : 20 - 85

(3 - 5) (5 - 5)

FOIL メソッドを適用する:

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

a = 3, b = - 5, c = 5, d = - 5

= 3 \cdot 5 + 3 (- 5) + (- 5) \cdot 5 + (- 5) (- 5)

マイナス・プラスの規則を適用する

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

= 3 \cdot 5 - 35 - 55 + 55

簡素化

3 \cdot 5 - 35 - 55 + 55 : 20 - 85

3 \cdot 5 - 35 - 55 + 55

類似した元を足す:

- 35 - 55 = - 85

= 3 \cdot 5 - 85 + 55

数を乗じる:

3 \cdot 5 = 15

= 15 - 85 + 55

累乗根の規則を適用する:

a a = a

55 = 5

= 15 - 85 + 5

数を足す:

15 + 5 = 20

= 20 - 85

= 20 - 85

= \frac{20 - 8 5}{2 ^{2}}

因数

20 - 85 : 4 (5 - 25)

20 - 85

書き換え

= 4 \cdot 5 - 4 \cdot 25

共通項をくくり出す

4

= 4 (5 - 25)

= \frac{4 ( 5 - 2 5 )}{2 ^{2}}

因数

4 : 2^{2}

因数

4 = 2^{2}

= \frac{2 ^{2} ( 5 - 2 5 )}{2 ^{2}}

共通因数を約分する:

2^{2}

= 5 - 25

= \frac{360 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4} + 240 ( - ( \frac{2 ( 5 - 1 ) 5 - 5}{4} ) ^{2} + 1 )}{1 - ( 5 - 2 5 )}

360 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4} = 902 (5 - 1) 5 - 5

360 \cdot \frac{2 ( 5 - 1 ) 5 - 5}{4}

分数を乗じる:

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

= \frac{2 ( 5 - 1 ) 5 - 5 \cdot 360}{4}

数を割る:

\frac{360}{4} = 90

= 902 (5 - 1) 5 - 5

240 1 - (\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2} = 240 (25 - 4)

240 1 - (\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2}

(\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2} = 5 - 25

(\frac{2 ( 5 - 1 ) 5 - 5}{4})^{2}

指数の規則を適用する:

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

= \frac{( 2 ( 5 - 1 ) 5 - 5 ) ^{2}}{4 ^{2}}

指数の規則を適用する:

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

(2 (5 - 1) 5 - 5)^{2} = (2)^{2} (5 - 5)^{2} (5 - 1)^{2}

= \frac{( 2 ) ^{2} ( 5 - 5 ) ^{2} ( 5 - 1 ) ^{2}}{4 ^{2}}

(2)^{2} : 2

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

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

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 2

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 ) ^{2}}{4 ^{2}}

(5 - 5)^{2} : 5 - 5

累乗根の規則を適用する:

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

= ((5 - 5)^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= (5 - 5)^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 5 - 5

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 )}{4 ^{2}}

因数

4^{2} : 2^{4}

因数

4 = 2^{2}

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

簡素化

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

(2^{2})^{2}

指数の規則を適用する:

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

= 2^{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= 2^{4}

= 2^{4}

= \frac{2 ( 5 - 1 ) ^{2} ( 5 - 5 )}{2 ^{4}}

共通因数を約分する:

2

= \frac{( 5 - 1 ) ^{2} ( 5 - 5 )}{2 ^{3}}

(5 - 1)^{2} = 6 - 25

(5 - 1)^{2}

完全平方式を適用する:

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

a = 5, b = 1

= (5)^{2} - 25 \cdot 1 + 1^{2}

簡素化

(5)^{2} - 25 \cdot 1 + 1^{2} : 6 - 25

(5)^{2} - 25 \cdot 1 + 1^{2}

規則を適用

1^{a} = 1

1^{2} = 1

= (5)^{2} - 2 \cdot 1 \cdot 5 + 1

(5)^{2} = 5

(5)^{2}

累乗根の規則を適用する:

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

= (5^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= 5^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 5

25 \cdot 1 = 25

25 \cdot 1

数を乗じる:

2 \cdot 1 = 2

= 25

= 5 - 25 + 1

数を足す:

5 + 1 = 6

= 6 - 25

= 6 - 25

= \frac{( 6 - 2 5 ) ( 5 - 5 )}{2 ^{3}}

因数

6 - 25 : 2 (3 - 5)

6 - 25

書き換え

= 2 \cdot 3 - 25

共通項をくくり出す

2

= 2 (3 - 5)

= \frac{2 ( 3 - 5 ) ( 5 - 5 )}{2 ^{3}}

共通因数を約分する:

2

= \frac{( 3 - 5 ) ( 5 - 5 )}{2 ^{2}}

2^{2} = 4

= \frac{( 3 - 5 ) ( 5 - 5 )}{4}

拡張

(3 - 5) (5 - 5) : 20 - 85

(3 - 5) (5 - 5)

FOIL メソッドを適用する:

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

a = 3, b = - 5, c = 5, d = - 5

= 3 \cdot 5 + 3 (- 5) + (- 5) \cdot 5 + (- 5) (- 5)

マイナス・プラスの規則を適用する

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

= 3 \cdot 5 - 35 - 55 + 55

簡素化

3 \cdot 5 - 35 - 55 + 55 : 20 - 85

3 \cdot 5 - 35 - 55 + 55

類似した元を足す:

- 35 - 55 = - 85

= 3 \cdot 5 - 85 + 55

数を乗じる:

3 \cdot 5 = 15

= 15 - 85 + 55

累乗根の規則を適用する:

a a = a

55 = 5

= 15 - 85 + 5

数を足す:

15 + 5 = 20

= 20 - 85

= 20 - 85

= \frac{20 - 8 5}{4}

因数

20 - 85 : 4 (5 - 25)

20 - 85

書き換え

= 4 \cdot 5 - 4 \cdot 25

共通項をくくり出す

4

= 4 (5 - 25)

= \frac{4 ( 5 - 2 5 )}{4}

数を割る:

\frac{4}{4} = 1

= 5 - 25

= 240 (- (5 - 25) + 1)

拡張

1 - (5 - 25) : 25 - 4

1 - (5 - 25)

- (5 - 25) : - 5 + 25

- (5 - 25)

括弧を分配する

= - (5) - (- 25)

マイナス・プラスの規則を適用する

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

= - 5 + 25

= 1 - 5 + 25

数を引く:

1 - 5 = - 4

= 25 - 4

= 240 (25 - 4)

= \frac{90 2 ( 5 - 1 ) 5 - 5 + 240 ( 2 5 - 4 )}{1 - ( 5 - 2 5 )}

拡張

1 - (5 - 25) : 25 - 4

1 - (5 - 25)

- (5 - 25) : - 5 + 25

- (5 - 25)

括弧を分配する

= - (5) - (- 25)

マイナス・プラスの規則を適用する

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

= - 5 + 25

= 1 - 5 + 25

数を引く:

1 - 5 = - 4

= 25 - 4

= \frac{90 2 ( 5 - 1 ) 5 - 5 + 240 ( 2 5 - 4 )}{2 5 - 4}

因数

902 (5 - 1) 5 - 5 + 240 (25 - 4) : 30 (32 (- 1 + 5) 5 - 5 + 16 (- 2 + 5))

902 (5 - 1) 5 - 5 + 240 (25 - 4)

書き換え

= 30 \cdot 32 (- 1 + 5) 5 - 5 + 30 \cdot 8 (- 4 + 25)

共通項をくくり出す

30

= 30 (32 (- 1 + 5) 5 - 5 + 8 (- 4 + 25))

因数

32 (5 - 1) 5 - 5 + 8 (25 - 4) : 32 (- 1 + 5) 5 - 5 + 16 (- 2 + 5)

32 (- 1 + 5) 5 - 5 + 8 (- 4 + 25)

8 (- 4 + 25) = 16 (- 2 + 5)

8 (- 4 + 25)

因数

- 4 + 25 : 2 (- 2 + 5)

- 4 + 25

書き換え

= - 2 \cdot 2 + 25

共通項をくくり出す

2

= 2 (- 2 + 5)

= 8 \cdot 2 (- 2 + 5)

改良

= 16 (- 2 + 5)

= 32 (5 - 1) 5 - 5 + 16 (5 - 2)

= 30 (32 (5 - 1) 5 - 5 + 16 (5 - 2))

= \frac{30 ( 3 2 ( - 1 + 5 ) 5 - 5 + 16 ( - 2 + 5 ) )}{2 5 - 4}

因数

25 - 4 : 2 (5 - 2)

25 - 4

書き換え

= 25 - 2 \cdot 2

共通項をくくり出す

2

= 2 (5 - 2)

= \frac{30 ( 3 2 ( - 1 + 5 ) 5 - 5 + 16 ( - 2 + 5 ) )}{2 ( 5 - 2 )}

数を割る:

\frac{30}{2} = 15

= \frac{15 ( 3 2 ( 5 - 1 ) 5 - 5 + 16 ( 5 - 2 ) )}{( 5 - 2 )}

括弧を削除する:

(a) = a

= \frac{15 ( 3 2 ( 5 - 1 ) 5 - 5 + 16 ( 5 - 2 ) )}{5 - 2}

有理化する

\frac{15 ( 3 2 ( 5 - 1 ) 5 - 5 + 16 ( 5 - 2 ) )}{5 - 2} : 1352 5 - 5 + 4510 5 - 5 + 240

\frac{15 ( 3 2 ( 5 - 1 ) 5 - 5 + 16 ( 5 - 2 ) )}{5 - 2}

共役で乗じる

\frac{5 + 2}{5 + 2}

= \frac{15 ( 3 2 ( 5 - 1 ) 5 - 5 + 16 ( 5 - 2 ) ) ( 5 + 2 )}{( 5 - 2 ) ( 5 + 2 )}

15 (32 (5 - 1) 5 - 5 + 16 (5 - 2)) (5 + 2) = 1352 5 - 5 + 4510 5 - 5 + 240

15 (32 (5 - 1) 5 - 5 + 16 (5 - 2)) (5 + 2)

拡張

(32 (5 - 1) 5 - 5 + 16 (5 - 2)) (5 + 2) : 92 5 - 5 + 310 5 - 5 + 16

(32 (5 - 1) 5 - 5 + 16 (5 - 2)) (5 + 2)

FOIL メソッドを適用する:

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

a = 32 (5 - 1) 5 - 5, b = 16 (5 - 2), c = 5, d = 2

= 32 (5 - 1) 5 - 5 5 + 32 (5 - 1) 5 - 5 \cdot 2 + 16 (5 - 2) 5 + 16 (5 - 2) \cdot 2

= 325 (5 - 1) 5 - 5 + 3 \cdot 22 (5 - 1) 5 - 5 + 165 (5 - 2) + 16 \cdot 2 (5 - 2)

簡素化

325 (5 - 1) 5 - 5 + 3 \cdot 22 (5 - 1) 5 - 5 + 165 (5 - 2) + 16 \cdot 2 (5 - 2) : 92 5 - 5 + 310 5 - 5 + 16

325 (5 - 1) 5 - 5 + 3 \cdot 22 (5 - 1) 5 - 5 + 165 (5 - 2) + 16 \cdot 2 (5 - 2)

325 (5 - 1) 5 - 5 = 310 (5 - 1) 5 - 5

325 (5 - 1) 5 - 5

累乗根の規則を適用する:

a b = a \cdot b

25 5 - 5 = 2 \cdot 5 (5 - 5)

= 3 (5 - 1) 2 \cdot 5 (5 - 5)

数を乗じる:

2 \cdot 5 = 10

= 3 (5 - 1) 10 (5 - 5)

累乗根の規則を適用する:

n ab = n a n b,

, 以下を想定

a \geq 0, b \geq 0

10 (5 - 5) = 10 5 - 5

= 310 (5 - 1) 5 - 5

3 \cdot 22 (5 - 1) 5 - 5 = 62 (5 - 1) 5 - 5

3 \cdot 22 (5 - 1) 5 - 5

数を乗じる:

3 \cdot 2 = 6

= 62 (5 - 1) 5 - 5

16 \cdot 2 (5 - 2) = 32 (5 - 2)

16 \cdot 2 (5 - 2)

数を乗じる:

16 \cdot 2 = 32

= 32 (5 - 2)

= 310 (5 - 1) 5 - 5 + 62 (5 - 1) 5 - 5 + 165 (5 - 2) + 32 (5 - 2)

= 310 (5 - 1) 5 - 5 + 62 (5 - 1) 5 - 5 + 165 (5 - 2) + 32 (5 - 2)

拡張

310 5 - 5 (5 - 1) : 152 5 - 5 - 310 5 - 5

310 5 - 5 (5 - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 310 5 - 5, b = 5, c = 1

= 310 5 - 5 5 - 310 5 - 5 \cdot 1

= 3105 5 - 5 - 3 \cdot 1 \cdot 10 5 - 5

簡素化

3105 5 - 5 - 3 \cdot 1 \cdot 10 5 - 5 : 152 5 - 5 - 310 5 - 5

3105 5 - 5 - 3 \cdot 1 \cdot 10 5 - 5

3105 5 - 5 = 152 5 - 5

3105 5 - 5

整数を因数分解する

10 = 5 \cdot 2

= 3 5 \cdot 2 5 5 - 5

累乗根の規則を適用する:

n ab = n a n b

5 \cdot 2 = 52

= 3525 5 - 5

累乗根の規則を適用する:

a a = a

55 = 5

= 3 \cdot 52 5 - 5

数を乗じる:

3 \cdot 5 = 15

= 152 5 - 5

3 \cdot 1 \cdot 10 5 - 5 = 310 5 - 5

3 \cdot 1 \cdot 10 5 - 5

数を乗じる:

3 \cdot 1 = 3

= 310 5 - 5

= 152 5 - 5 - 310 5 - 5

= 152 5 - 5 - 310 5 - 5

= 152 5 - 5 - 310 5 - 5 + 62 (5 - 1) 5 - 5 + 165 (5 - 2) + 32 (5 - 2)

拡張

62 5 - 5 (5 - 1) : 610 5 - 5 - 62 5 - 5

62 5 - 5 (5 - 1)

分配法則を適用する:

a (b - c) = ab - a c

a = 62 5 - 5, b = 5, c = 1

= 62 5 - 5 5 - 62 5 - 5 \cdot 1

= 625 5 - 5 - 6 \cdot 1 \cdot 2 5 - 5

簡素化

625 5 - 5 - 6 \cdot 1 \cdot 2 5 - 5 : 610 5 - 5 - 62 5 - 5

625 5 - 5 - 6 \cdot 1 \cdot 2 5 - 5

625 5 - 5 = 610 5 - 5

625 5 - 5

累乗根の規則を適用する:

a b = a \cdot b

25 5 - 5 = 2 \cdot 5 (5 - 5)

= 6 2 \cdot 5 (5 - 5)

数を乗じる:

2 \cdot 5 = 10

= 6 10 (5 - 5)

累乗根の規則を適用する:

n ab = n a n b,

, 以下を想定

a \geq 0, b \geq 0

10 (5 - 5) = 10 5 - 5

= 610 5 - 5

6 \cdot 1 \cdot 2 5 - 5 = 62 5 - 5

6 \cdot 1 \cdot 2 5 - 5

数を乗じる:

6 \cdot 1 = 6

= 62 5 - 5

= 610 5 - 5 - 62 5 - 5

= 610 5 - 5 - 62 5 - 5

= 152 5 - 5 - 310 5 - 5 + 610 5 - 5 - 62 5 - 5 + 165 (5 - 2) + 32 (5 - 2)

拡張

165 (5 - 2) : 80 - 325

165 (5 - 2)

分配法則を適用する:

a (b - c) = ab - a c

a = 165, b = 5, c = 2

= 1655 - 165 \cdot 2

= 1655 - 16 \cdot 25

簡素化

1655 - 16 \cdot 25 : 80 - 325

1655 - 16 \cdot 25

1655 = 80

1655

累乗根の規則を適用する:

a a = a

55 = 5

= 16 \cdot 5

数を乗じる:

16 \cdot 5 = 80

= 80

16 \cdot 25 = 325

16 \cdot 25

数を乗じる:

16 \cdot 2 = 32

= 325

= 80 - 325

= 80 - 325

= 152 5 - 5 - 310 5 - 5 + 610 5 - 5 - 62 5 - 5 + 80 - 325 + 32 (5 - 2)

拡張

32 (5 - 2) : 325 - 64

32 (5 - 2)

分配法則を適用する:

a (b - c) = ab - a c

a = 32, b = 5, c = 2

= 325 - 32 \cdot 2

数を乗じる:

32 \cdot 2 = 64

= 325 - 64

= 152 5 - 5 - 310 5 - 5 + 610 5 - 5 - 62 5 - 5 + 80 - 325 + 325 - 64

簡素化

152 5 - 5 - 310 5 - 5 + 610 5 - 5 - 62 5 - 5 + 80 - 325 + 325 - 64 : 92 5 - 5 + 310 5 - 5 + 16

152 5 - 5 - 310 5 - 5 + 610 5 - 5 - 62 5 - 5 + 80 - 325 + 325 - 64

類似した元を足す:

- 310 5 - 5 + 610 5 - 5 = 310 5 - 5

= 152 5 - 5 + 310 5 - 5 - 62 5 - 5 + 80 - 325 + 325 - 64

類似した元を足す:

152 5 - 5 - 62 5 - 5 = 92 5 - 5

= 92 5 - 5 + 310 5 - 5 + 80 - 325 + 325 - 64

類似した元を足す:

- 325 + 325 = 0

= 92 5 - 5 + 310 5 - 5 + 80 - 64

数を足す/引く:

80 - 64 = 16

= 92 5 - 5 + 310 5 - 5 + 16

= 92 5 - 5 + 310 5 - 5 + 16

= 92 5 - 5 + 310 5 - 5 + 16

= 15 (92 5 - 5 + 310 5 - 5 + 16)

拡張

15 (92 5 - 5 + 310 5 - 5 + 16) : 1352 5 - 5 + 4510 5 - 5 + 240

15 (92 5 - 5 + 310 5 - 5 + 16)

括弧を分配する

= 15 \cdot 92 5 - 5 + 15 \cdot 310 5 - 5 + 15 \cdot 16

簡素化

15 \cdot 92 5 - 5 + 15 \cdot 310 5 - 5 + 15 \cdot 16 : 1352 5 - 5 + 4510 5 - 5 + 240

15 \cdot 92 5 - 5 + 15 \cdot 310 5 - 5 + 15 \cdot 16

数を乗じる:

15 \cdot 9 = 135

= 1352 5 - 5 + 15 \cdot 310 5 - 5 + 15 \cdot 16

数を乗じる:

15 \cdot 3 = 45

= 1352 5 - 5 + 4510 5 - 5 + 15 \cdot 16

数を乗じる:

15 \cdot 16 = 240

= 1352 5 - 5 + 4510 5 - 5 + 240

= 1352 5 - 5 + 4510 5 - 5 + 240

= 1352 5 - 5 + 4510 5 - 5 + 240

(5 - 2) (5 + 2) = 1

(5 - 2) (5 + 2)

2乗の差の公式を適用する:

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

a = 5, b = 2

= (5)^{2} - 2^{2}

簡素化

(5)^{2} - 2^{2} : 1

(5)^{2} - 2^{2}

(5)^{2} = 5

(5)^{2}

累乗根の規則を適用する:

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

= (5^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= 5^{\frac{1}{2} \cdot 2}

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

\frac{1}{2} \cdot 2

分数を乗じる:

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

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

共通因数を約分する:

2

= 1

= 5

2^{2} = 4

2^{2}

2^{2} = 4

= 4

= 5 - 4

数を引く:

5 - 4 = 1

= 1

= 1

= \frac{135 2 5 - 5 + 45 10 5 - 5 + 240}{1}

規則を適用

\frac{a}{1} = a

= 1352 5 - 5 + 4510 5 - 5 + 240

= 1352 5 - 5 + 4510 5 - 5 + 240

= 1352 5 - 5 + 4510 5 - 5 + 240

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