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人気のある三角関数 >

arcsin(6x)+arcsin(6sqrt(3)x)=-pi/2

解

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

解

x = - \frac{1}{12}

解答ステップ

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

三角関数の公式を使用して書き換える

arcsin (6 x) + arcsin (63 x)

和・積の公式を使用する:

arcsin (s) + arcsin (t) = arcsin (s 1 - t^{2} + t 1 - s^{2})

= arcsin (6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2})

arcsin (6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2}) = - \frac{π}{2}

三角関数の逆数プロパティを適用する

arcsin (6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2}) = - \frac{π}{2}

arcsin (x) = a \Rightarrow x = sin (a)

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = sin (- \frac{π}{2})

sin (- \frac{π}{2}) = - 1

sin (- \frac{π}{2})

次のプロパティを使用する:

sin (- x) = - sin (x)

sin (- \frac{π}{2}) = - sin (\frac{π}{2})

= - sin (\frac{π}{2})

次の自明恒等式を使用する:

sin (\frac{π}{2}) = 1

sin (\frac{π}{2})

sin (x)

2 πn

循環を含む周期性テーブル:

x 0 \frac{π}{6} \frac{π}{4} \frac{π}{3} \frac{π}{2} \frac{2 π}{3} \frac{3 π}{4} \frac{5 π}{6} sin (x) 0 \frac{1}{2} \frac{2}{2} \frac{3}{2} 1 \frac{3}{2} \frac{2}{2} \frac{1}{2} x π \frac{7 π}{6} \frac{5 π}{4} \frac{4 π}{3} \frac{3 π}{2} \frac{5 π}{3} \frac{7 π}{4} \frac{11 π}{6} sin (x) 0 - \frac{1}{2} - \frac{2}{2} - \frac{3}{2} - 1 - \frac{3}{2} - \frac{2}{2} - \frac{1}{2}

= 1

= - 1

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

解く

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1 : x = - \frac{1}{12}

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

平方根を削除する

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

両辺から

63 x 1 - (6 x)^{2}

を引く

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} - 63 x 1 - (6 x)^{2} = - 1 - 63 x 1 - (6 x)^{2}

簡素化

6 1 - (63 x)^{2} x = - 1 - 63 x 1 - (6 x)^{2}

両辺を2乗する:

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

(6 1 - (63 x)^{2} x)^{2} = (- 1 - 63 x 1 - (6 x)^{2})^{2}

拡張

(6 1 - (63 x)^{2} x)^{2} : 36 x^{2} - 3888 x^{4}

(6 1 - (63 x)^{2} x)^{2}

指数の規則を適用する:

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

= 6^{2} x^{2} (1 - (63 x)^{2})^{2}

(1 - (63 x)^{2})^{2} : 1 - (63 x)^{2}

累乗根の規則を適用する:

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

= ((1 - (63 x)^{2})^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= (1 - (63 x)^{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

= 1 - (63 x)^{2}

= 6^{2} (1 - (63 x)^{2}) x^{2}

6^{2} = 36

= 36 (1 - (63 x)^{2}) x^{2}

拡張

36 (1 - (63 x)^{2}) x^{2} : 36 x^{2} - 3888 x^{4}

36 (1 - (63 x)^{2}) x^{2}

(63 x)^{2} = 6^{2} \cdot 3 x^{2}

(63 x)^{2}

指数の規則を適用する:

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

= 6^{2} (3)^{2} x^{2}

(3)^{2} : 3

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

= 3^{\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

= 3

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

= 36 x^{2} (- 6^{2} \cdot 3 x^{2} + 1)

6^{2} \cdot 3 x^{2} = 108 x^{2}

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

6^{2} = 36

= 36 \cdot 3 x^{2}

数を乗じる:

36 \cdot 3 = 108

= 108 x^{2}

= 36 x^{2} (- 108 x^{2} + 1)

= 36 x^{2} (1 - 108 x^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 36 x^{2}, b = 1, c = 108 x^{2}

= 36 x^{2} \cdot 1 - 36 x^{2} \cdot 108 x^{2}

= 36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x^{2}

簡素化

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x^{2} : 36 x^{2} - 3888 x^{4}

36 \cdot 1 \cdot x^{2} - 36 \cdot 108 x^{2} x^{2}

36 \cdot 1 \cdot x^{2} = 36 x^{2}

36 \cdot 1 \cdot x^{2}

数を乗じる:

36 \cdot 1 = 36

= 36 x^{2}

36 \cdot 108 x^{2} x^{2} = 3888 x^{4}

36 \cdot 108 x^{2} x^{2}

数を乗じる:

36 \cdot 108 = 3888

= 3888 x^{2} x^{2}

指数の規則を適用する:

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

x^{2} x^{2} = x^{2 + 2}

= 3888 x^{2 + 2}

数を足す:

2 + 2 = 4

= 3888 x^{4}

= 36 x^{2} - 3888 x^{4}

= 36 x^{2} - 3888 x^{4}

= 36 x^{2} - 3888 x^{4}

拡張

(- 1 - 63 x 1 - (6 x)^{2})^{2} : 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

(- 1 - 63 x 1 - (6 x)^{2})^{2}

= (- 1 - 63 1 - (6 x)^{2} x)^{2}

完全平方式を適用する:

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

a = - 1, b = 63 x 1 - (6 x)^{2}

= (- 1)^{2} - 2 (- 1) \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{2}

簡素化

(- 1)^{2} - 2 (- 1) \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{2} : 1 + 123 1 - (6 x)^{2} x + 1081 - (6 x)^{2} x^{2}

(- 1)^{2} - 2 (- 1) \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{2}

規則を適用

- (- a) = a

= (- 1)^{2} + 2 \cdot 1 \cdot 63 x 1 - (6 x)^{2} + (63 x 1 - (6 x)^{2})^{2}

(- 1)^{2} = 1

(- 1)^{2}

指数の規則を適用する:

n

が偶数であれば

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

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

= 1^{2}

規則を適用

1^{a} = 1

= 1

2 \cdot 1 \cdot 63 x 1 - (6 x)^{2} = 123 1 - (6 x)^{2} x

2 \cdot 1 \cdot 63 x 1 - (6 x)^{2}

数を乗じる:

2 \cdot 1 \cdot 6 = 12

= 123 1 - (6 x)^{2} x

(63 x 1 - (6 x)^{2})^{2} = 1081 - (6 x)^{2} x^{2}

(63 x 1 - (6 x)^{2})^{2}

指数の規則を適用する:

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

= 6^{2} (3)^{2} x^{2} (1 - (6 x)^{2})^{2}

(3)^{2} : 3

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

= 3^{\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

= 3

= 6^{2} \cdot 3 x^{2} (1 - (6 x)^{2})^{2}

(1 - (6 x)^{2})^{2} : 1 - (6 x)^{2}

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

= (1 - (6 x)^{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

= 1 - (6 x)^{2}

= 6^{2} \cdot 3 x^{2} (1 - (6 x)^{2})

改良

= 108 (1 - (6 x)^{2}) x^{2}

= 1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2}

= 1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2}

拡張

1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2} : 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

1 + 123 1 - (6 x)^{2} x + 108 (1 - (6 x)^{2}) x^{2}

1 - (6 x)^{2} = 1 - 36 x^{2}

1 - (6 x)^{2}

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

指数の規則を適用する:

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

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 1 - 36 x^{2}

= 1 + 123 x - 36 x^{2} + 1 + 108 x^{2} (- (6 x)^{2} + 1)

(6 x)^{2} = 36 x^{2}

(6 x)^{2}

指数の規則を適用する:

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

= 6^{2} x^{2}

6^{2} = 36

= 36 x^{2}

= 1 + 123 x - 36 x^{2} + 1 + 108 x^{2} (- 36 x^{2} + 1)

= 1 + 123 x 1 - 36 x^{2} + 108 x^{2} (1 - 36 x^{2})

拡張

108 x^{2} (1 - 36 x^{2}) : 108 x^{2} - 3888 x^{4}

108 x^{2} (1 - 36 x^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 108 x^{2}, b = 1, c = 36 x^{2}

= 108 x^{2} \cdot 1 - 108 x^{2} \cdot 36 x^{2}

= 108 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x^{2}

簡素化

108 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x^{2} : 108 x^{2} - 3888 x^{4}

108 \cdot 1 \cdot x^{2} - 108 \cdot 36 x^{2} x^{2}

108 \cdot 1 \cdot x^{2} = 108 x^{2}

108 \cdot 1 \cdot x^{2}

数を乗じる:

108 \cdot 1 = 108

= 108 x^{2}

108 \cdot 36 x^{2} x^{2} = 3888 x^{4}

108 \cdot 36 x^{2} x^{2}

数を乗じる:

108 \cdot 36 = 3888

= 3888 x^{2} x^{2}

指数の規則を適用する:

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

x^{2} x^{2} = x^{2 + 2}

= 3888 x^{2 + 2}

数を足す:

2 + 2 = 4

= 3888 x^{4}

= 108 x^{2} - 3888 x^{4}

= 108 x^{2} - 3888 x^{4}

= 1 + 123 1 - 36 x^{2} x + 108 x^{2} - 3888 x^{4}

= 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

= 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

両辺から

108 x^{2} - 3888 x^{4}

を引く

36 x^{2} - 3888 x^{4} - (108 x^{2} - 3888 x^{4}) = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4} - (108 x^{2} - 3888 x^{4})

簡素化

- 72 x^{2} = 123 1 - 36 x^{2} x + 1

両辺から

1

を引く

- 72 x^{2} - 1 = 123 1 - 36 x^{2} x + 1 - 1

簡素化

- 72 x^{2} - 1 = 123 1 - 36 x^{2} x

両辺を2乗する:

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

36 x^{2} - 3888 x^{4} = 1 + 123 x 1 - 36 x^{2} + 108 x^{2} - 3888 x^{4}

(- 72 x^{2} - 1)^{2} = (123 1 - 36 x^{2} x)^{2}

拡張

(- 72 x^{2} - 1)^{2} : 5184 x^{4} + 144 x^{2} + 1

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

完全平方式を適用する:

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

a = - 72 x^{2}, b = 1

= (- 72 x^{2})^{2} - 2 (- 72 x^{2}) \cdot 1 + 1^{2}

簡素化

(- 72 x^{2})^{2} - 2 (- 72 x^{2}) \cdot 1 + 1^{2} : 5184 x^{4} + 144 x^{2} + 1

(- 72 x^{2})^{2} - 2 (- 72 x^{2}) \cdot 1 + 1^{2}

規則を適用

1^{a} = 1

1^{2} = 1

= (- 72 x^{2})^{2} - 2 \cdot 1 \cdot (- 72 x^{2}) + 1

規則を適用

- (- a) = a

= (- 72 x^{2})^{2} + 2 \cdot 72 x^{2} \cdot 1 + 1

(- 72 x^{2})^{2} = 5184 x^{4}

(- 72 x^{2})^{2}

指数の規則を適用する:

n

が偶数であれば

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

(- 72 x^{2})^{2} = (72 x^{2})^{2}

= (72 x^{2})^{2}

指数の規則を適用する:

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

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

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

指数の規則を適用する:

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

= x^{2 \cdot 2}

数を乗じる:

2 \cdot 2 = 4

= x^{4}

= 7 2^{2} x^{4}

7 2^{2} = 5184

= 5184 x^{4}

2 \cdot 72 x^{2} \cdot 1 = 144 x^{2}

2 \cdot 72 x^{2} \cdot 1

数を乗じる:

2 \cdot 72 \cdot 1 = 144

= 144 x^{2}

= 5184 x^{4} + 144 x^{2} + 1

= 5184 x^{4} + 144 x^{2} + 1

拡張

(123 1 - 36 x^{2} x)^{2} : 432 x^{2} - 15552 x^{4}

(123 1 - 36 x^{2} x)^{2}

指数の規則を適用する:

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

= 1 2^{2} (3)^{2} x^{2} (1 - 36 x^{2})^{2}

(3)^{2} : 3

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

= 3^{\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

= 3

= 1 2^{2} \cdot 3 (1 - 36 x^{2})^{2} x^{2}

(1 - 36 x^{2})^{2} : 1 - 36 x^{2}

累乗根の規則を適用する:

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

= ((1 - 36 x^{2})^{\frac{1}{2}})^{2}

指数の規則を適用する:

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

= (1 - 36 x^{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

= 1 - 36 x^{2}

= 1 2^{2} \cdot 3 (1 - 36 x^{2}) x^{2}

改良

= 432 (1 - 36 x^{2}) x^{2}

拡張

432 (1 - 36 x^{2}) x^{2} : 432 x^{2} - 15552 x^{4}

432 (1 - 36 x^{2}) x^{2}

= 432 x^{2} (1 - 36 x^{2})

分配法則を適用する:

a (b - c) = ab - a c

a = 432 x^{2}, b = 1, c = 36 x^{2}

= 432 x^{2} \cdot 1 - 432 x^{2} \cdot 36 x^{2}

= 432 \cdot 1 \cdot x^{2} - 432 \cdot 36 x^{2} x^{2}

簡素化

432 \cdot 1 \cdot x^{2} - 432 \cdot 36 x^{2} x^{2} : 432 x^{2} - 15552 x^{4}

432 \cdot 1 \cdot x^{2} - 432 \cdot 36 x^{2} x^{2}

432 \cdot 1 \cdot x^{2} = 432 x^{2}

432 \cdot 1 \cdot x^{2}

数を乗じる:

432 \cdot 1 = 432

= 432 x^{2}

432 \cdot 36 x^{2} x^{2} = 15552 x^{4}

432 \cdot 36 x^{2} x^{2}

数を乗じる:

432 \cdot 36 = 15552

= 15552 x^{2} x^{2}

指数の規則を適用する:

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

x^{2} x^{2} = x^{2 + 2}

= 15552 x^{2 + 2}

数を足す:

2 + 2 = 4

= 15552 x^{4}

= 432 x^{2} - 15552 x^{4}

= 432 x^{2} - 15552 x^{4}

= 432 x^{2} - 15552 x^{4}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

解く

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4} : x = \frac{1}{12}, x = - \frac{1}{12}

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

15552 x^{4}

を左側に移動します

5184 x^{4} + 144 x^{2} + 1 = 432 x^{2} - 15552 x^{4}

両辺に

15552 x^{4}

を足す

5184 x^{4} + 144 x^{2} + 1 + 15552 x^{4} = 432 x^{2} - 15552 x^{4} + 15552 x^{4}

簡素化

20736 x^{4} + 144 x^{2} + 1 = 432 x^{2}

20736 x^{4} + 144 x^{2} + 1 = 432 x^{2}

432 x^{2}

を左側に移動します

20736 x^{4} + 144 x^{2} + 1 = 432 x^{2}

両辺から

432 x^{2}

を引く

20736 x^{4} + 144 x^{2} + 1 - 432 x^{2} = 432 x^{2} - 432 x^{2}

簡素化

20736 x^{4} - 288 x^{2} + 1 = 0

20736 x^{4} - 288 x^{2} + 1 = 0

以下で両辺を割る

20736

\frac{20736 x ^{4}}{20736} - \frac{288 x ^{2}}{20736} + \frac{1}{20736} = \frac{0}{20736}

標準的な形式で書く

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

x^{4} - \frac{x ^{2}}{72} + \frac{1}{20736} = 0

equationを

u = x^{2}

と以下で書き換える:

u^{2} = x^{4}

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

解く

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0 : u = \frac{1}{144}

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

以下の最小公倍数を求める:

72, 20736 : 20736

72, 20736

最小公倍数 (LCM)

以下の素因数分解:

72 : 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3

72

72272 = 36 \cdot 2

で割る

= 2 \cdot 36

36236 = 18 \cdot 2

で割る

= 2 \cdot 2 \cdot 18

18218 = 9 \cdot 2

で割る

= 2 \cdot 2 \cdot 2 \cdot 9

939 = 3 \cdot 3

で割る

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

2, 3

はすべて素数である。ゆえにさらに因数分解することはできない

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

以下の素因数分解:

20736 : 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3

20736

20736220736 = 10368 \cdot 2

で割る

= 2 \cdot 10368

10368210368 = 5184 \cdot 2

で割る

= 2 \cdot 2 \cdot 5184

518425184 = 2592 \cdot 2

で割る

= 2 \cdot 2 \cdot 2 \cdot 2592

259222592 = 1296 \cdot 2

で割る

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 1296

129621296 = 648 \cdot 2

で割る

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 648

6482648 = 324 \cdot 2

で割る

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 324

3242324 = 162 \cdot 2

で割る

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 162

1622162 = 81 \cdot 2

で割る

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 81

81381 = 27 \cdot 3

で割る

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 27

27327 = 9 \cdot 3

で割る

= 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 9

939 = 3 \cdot 3

で割る

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

2, 3

はすべて素数である。ゆえにさらに因数分解することはできない

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

72

または以下のいずれかで生じる最大回数, 各因数を乗じる:

20736

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

数を乗じる:

2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 2 \cdot 3 \cdot 3 \cdot 3 \cdot 3 = 20736

= 20736

以下で乗じる: LCM=

20736

u^{2} \cdot 20736 - \frac{u}{72} \cdot 20736 + \frac{1}{20736} \cdot 20736 = 0 \cdot 20736

簡素化

20736 u^{2} - 288 u + 1 = 0

以下で両辺を割る

20736

\frac{20736 u ^{2}}{20736} - \frac{288 u}{20736} + \frac{1}{20736} = \frac{0}{20736}

標準的な形式で書く

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

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

解くとthe二次式

u^{2} - \frac{u}{72} + \frac{1}{20736} = 0

二次Equationの公式:

次の場合:

a = 1, b = - \frac{1}{72}, c = \frac{1}{20736}

u_{1, 2} = \frac{- ( - \frac{1}{72} ) \pm ( - \frac{1}{72} ) ^{2} - 4 \cdot 1 \cdot \frac{1}{20736}}{2 \cdot 1}

u_{1, 2} = \frac{- ( - \frac{1}{72} ) \pm ( - \frac{1}{72} ) ^{2} - 4 \cdot 1 \cdot \frac{1}{20736}}{2 \cdot 1}

(- \frac{1}{72})^{2} - 4 \cdot 1 \cdot \frac{1}{20736} = 0

(- \frac{1}{72})^{2} - 4 \cdot 1 \cdot \frac{1}{20736}

(- \frac{1}{72})^{2} = \frac{1}{7 2 ^{2}}

(- \frac{1}{72})^{2}

指数の規則を適用する:

n

が偶数であれば

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

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

= (\frac{1}{72})^{2}

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

4 \cdot 1 \cdot \frac{1}{20736} = \frac{1}{5184}

4 \cdot 1 \cdot \frac{1}{20736}

分数を乗じる:

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

= 1 \cdot \frac{1 \cdot 4}{20736}

\frac{1 \cdot 4}{20736} = \frac{1}{5184}

\frac{1 \cdot 4}{20736}

数を乗じる:

1 \cdot 4 = 4

= \frac{4}{20736}

共通因数を約分する:

4

= \frac{1}{5184}

= 1 \cdot \frac{1}{5184}

乗算:

1 \cdot \frac{1}{5184} = \frac{1}{5184}

= \frac{1}{5184}

= \frac{1}{7 2 ^{2}} - \frac{1}{5184}

7 2^{2} = 5184

= \frac{1}{5184} - \frac{1}{5184}

類似した元を足す:

\frac{1}{5184} - \frac{1}{5184} = 0

= 0

u_{1, 2} = \frac{- ( - \frac{1}{72} ) \pm 0}{2 \cdot 1}

u = \frac{- ( - \frac{1}{72} )}{2 \cdot 1}

\frac{- ( - \frac{1}{72} )}{2 \cdot 1} = \frac{1}{144}

\frac{- ( - \frac{1}{72} )}{2 \cdot 1}

規則を適用

- (- a) = a

= \frac{\frac{1}{72}}{2 \cdot 1}

数を乗じる:

2 \cdot 1 = 2

= \frac{\frac{1}{72}}{2}

分数の規則を適用する:

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

= \frac{1}{72 \cdot 2}

数を乗じる:

72 \cdot 2 = 144

= \frac{1}{144}

u = \frac{1}{144}

二次equationの解:

u = \frac{1}{144}

u = \frac{1}{144}

再び

u = x^{2}

に置き換えて以下を解く:

x

解く

x^{2} = \frac{1}{144} : x = \frac{1}{12}, x = - \frac{1}{12}

x^{2} = \frac{1}{144}

x^{2} = f (a)

の場合, 解は

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

x = \frac{1}{144}, x = - \frac{1}{144}

\frac{1}{144} = \frac{1}{12}

\frac{1}{144}

累乗根の規則を適用する:

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

= \frac{1}{144}

累乗根の規則を適用する:

1 = 1

1 = 1

= \frac{1}{144}

144 = 12

144

数を因数に分解する:

144 = 1 2^{2}

= 1 2^{2}

累乗根の規則を適用する:

a^{2} = a, a \geq 0

1 2^{2} = 12

= 12

= \frac{1}{12}

- \frac{1}{144} = - \frac{1}{12}

- \frac{1}{144}

累乗根の規則を適用する:

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

= - \frac{1}{144}

累乗根の規則を適用する:

1 = 1

1 = 1

= - \frac{1}{144}

144 = 12

144

数を因数に分解する:

144 = 1 2^{2}

= 1 2^{2}

累乗根の規則を適用する:

a^{2} = a, a \geq 0

1 2^{2} = 12

= 12

= - \frac{1}{12}

x = \frac{1}{12}, x = - \frac{1}{12}

解答は

x = \frac{1}{12}, x = - \frac{1}{12}

x = \frac{1}{12}, x = - \frac{1}{12}

解を検算する:

x = \frac{1}{12}

偽

, x = - \frac{1}{12}

真

6 x 1 - (63 x)^{2} + 63 x 1 - (6 x)^{2} = - 1

に当てはめて解を確認する

equationに一致しないものを削除する。

挿入

x = \frac{1}{12} :

偽

6 (\frac{1}{12}) 1 - (63 (\frac{1}{12}))^{2} + 63 (\frac{1}{12}) 1 - (6 (\frac{1}{12}))^{2} = - 1

6 (\frac{1}{12}) 1 - (63 (\frac{1}{12}))^{2} + 63 (\frac{1}{12}) 1 - (6 (\frac{1}{12}))^{2} = 1

6 (\frac{1}{12}) 1 - (63 (\frac{1}{12}))^{2} + 63 (\frac{1}{12}) 1 - (6 (\frac{1}{12}))^{2}

括弧を削除する:

(a) = a

= 6 \cdot \frac{1}{12} 1 - (63 \frac{1}{12})^{2} + 63 \frac{1}{12} 1 - (6 \cdot \frac{1}{12})^{2}

6 \cdot \frac{1}{12} 1 - (63 \frac{1}{12})^{2} = \frac{1}{4}

6 \cdot \frac{1}{12} 1 - (63 \frac{1}{12})^{2}

1 - (63 \frac{1}{12})^{2} = \frac{1}{2}

1 - (63 \frac{1}{12})^{2}

(63 \frac{1}{12})^{2} = \frac{3}{4}

(63 \frac{1}{12})^{2}

乗じる

63 \frac{1}{12} : \frac{3}{2}

63 \frac{1}{12}

分数を乗じる:

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

= \frac{1 \cdot 6 3}{12}

数を乗じる:

1 \cdot 6 = 6

= \frac{6 3}{12}

共通因数を約分する:

6

= \frac{3}{2}

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

指数の規則を適用する:

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

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

(3)^{2} : 3

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

= 3^{\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

= 3

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

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

結合

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

1 - \frac{3}{4}

元を分数に変換する:

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

= \frac{1 \cdot 4}{4} - \frac{3}{4}

分母が等しいので, 分数を組み合わせる:

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

= \frac{1 \cdot 4 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

数を乗じる:

1 \cdot 4 = 4

= 4 - 3

数を引く:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

累乗根の規則を適用する:

, 以下を想定

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

2^{2} = 2

= 2

= \frac{1}{2}

規則を適用

1 = 1

= \frac{1}{2}

= 6 \cdot \frac{1}{2} \cdot \frac{1}{12}

分数を乗じる:

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

= \frac{1 \cdot 1 \cdot 6}{12 \cdot 2}

数を乗じる:

1 \cdot 1 \cdot 6 = 6

= \frac{6}{12 \cdot 2}

数を乗じる:

12 \cdot 2 = 24

= \frac{6}{24}

共通因数を約分する:

6

= \frac{1}{4}

63 \frac{1}{12} 1 - (6 \cdot \frac{1}{12})^{2} = \frac{3}{4}

63 \frac{1}{12} 1 - (6 \cdot \frac{1}{12})^{2}

1 - (6 \cdot \frac{1}{12})^{2} = \frac{3}{2}

1 - (6 \cdot \frac{1}{12})^{2}

(6 \cdot \frac{1}{12})^{2} = \frac{1}{4}

(6 \cdot \frac{1}{12})^{2}

乗じる

6 \cdot \frac{1}{12} : \frac{1}{2}

6 \cdot \frac{1}{12}

分数を乗じる:

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

= \frac{1 \cdot 6}{12}

数を乗じる:

1 \cdot 6 = 6

= \frac{6}{12}

共通因数を約分する:

6

= \frac{1}{2}

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

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

結合

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

1 - \frac{1}{4}

元を分数に変換する:

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

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

分母が等しいので, 分数を組み合わせる:

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

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

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

数を乗じる:

1 \cdot 4 = 4

= 4 - 1

数を引く:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

累乗根の規則を適用する:

, 以下を想定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

2^{2} = 2

= 2

= \frac{3}{2}

= 63 \frac{1}{12} \cdot \frac{3}{2}

分数を乗じる:

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

= \frac{1 \cdot 3 \cdot 6 3}{12 \cdot 2}

1 \cdot 3 \cdot 63 = 18

1 \cdot 3 \cdot 63

数を乗じる:

1 \cdot 6 = 6

= 633

累乗根の規則を適用する:

a a = a

33 = 3

= 6 \cdot 3

数を乗じる:

6 \cdot 3 = 18

= 18

= \frac{18}{12 \cdot 2}

数を乗じる:

12 \cdot 2 = 24

= \frac{18}{24}

共通因数を約分する:

6

= \frac{3}{4}

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

規則を適用

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

= \frac{1 + 3}{4}

数を足す:

1 + 3 = 4

= \frac{4}{4}

規則を適用

\frac{a}{a} = 1

= 1

1 = - 1

偽

挿入

x = - \frac{1}{12} :

真

6 (- \frac{1}{12}) 1 - (63 (- \frac{1}{12}))^{2} + 63 (- \frac{1}{12}) 1 - (6 (- \frac{1}{12}))^{2} = - 1

6 (- \frac{1}{12}) 1 - (63 (- \frac{1}{12}))^{2} + 63 (- \frac{1}{12}) 1 - (6 (- \frac{1}{12}))^{2} = - 1

6 (- \frac{1}{12}) 1 - (63 (- \frac{1}{12}))^{2} + 63 (- \frac{1}{12}) 1 - (6 (- \frac{1}{12}))^{2}

括弧を削除する:

(- a) = - a

= - 6 \cdot \frac{1}{12} 1 - (- 63 \frac{1}{12})^{2} - 63 \frac{1}{12} 1 - (- 6 \cdot \frac{1}{12})^{2}

6 \cdot \frac{1}{12} 1 - (- 63 \frac{1}{12})^{2} = \frac{1}{4}

6 \cdot \frac{1}{12} 1 - (- 63 \frac{1}{12})^{2}

1 - (- 63 \frac{1}{12})^{2} = \frac{1}{2}

1 - (- 63 \frac{1}{12})^{2}

(- 63 \frac{1}{12})^{2} = \frac{3}{4}

(- 63 \frac{1}{12})^{2}

乗じる

- 63 \frac{1}{12} : - \frac{3}{2}

- 63 \frac{1}{12}

分数を乗じる:

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

= - \frac{1 \cdot 6 3}{12}

数を乗じる:

1 \cdot 6 = 6

= - \frac{6 3}{12}

共通因数を約分する:

6

= - \frac{3}{2}

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

指数の規則を適用する:

n

が偶数であれば

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

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

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

指数の規則を適用する:

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

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

(3)^{2} : 3

累乗根の規則を適用する:

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

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

指数の規則を適用する:

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

= 3^{\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

= 3

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

2^{2} = 4

= \frac{3}{4}

= 1 - \frac{3}{4}

結合

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

1 - \frac{3}{4}

元を分数に変換する:

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

= \frac{1 \cdot 4}{4} - \frac{3}{4}

分母が等しいので, 分数を組み合わせる:

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

= \frac{1 \cdot 4 - 3}{4}

1 \cdot 4 - 3 = 1

1 \cdot 4 - 3

数を乗じる:

1 \cdot 4 = 4

= 4 - 3

数を引く:

4 - 3 = 1

= 1

= \frac{1}{4}

= \frac{1}{4}

累乗根の規則を適用する:

, 以下を想定

a \geq 0, b \geq 0

= \frac{1}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

2^{2} = 2

= 2

= \frac{1}{2}

規則を適用

1 = 1

= \frac{1}{2}

= 6 \cdot \frac{1}{2} \cdot \frac{1}{12}

分数を乗じる:

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

= \frac{1 \cdot 1 \cdot 6}{12 \cdot 2}

数を乗じる:

1 \cdot 1 \cdot 6 = 6

= \frac{6}{12 \cdot 2}

数を乗じる:

12 \cdot 2 = 24

= \frac{6}{24}

共通因数を約分する:

6

= \frac{1}{4}

63 \frac{1}{12} 1 - (- 6 \cdot \frac{1}{12})^{2} = \frac{3}{4}

63 \frac{1}{12} 1 - (- 6 \cdot \frac{1}{12})^{2}

1 - (- 6 \cdot \frac{1}{12})^{2} = \frac{3}{2}

1 - (- 6 \cdot \frac{1}{12})^{2}

(- 6 \cdot \frac{1}{12})^{2} = \frac{1}{4}

(- 6 \cdot \frac{1}{12})^{2}

乗じる

- 6 \cdot \frac{1}{12} : - \frac{1}{2}

- 6 \cdot \frac{1}{12}

分数を乗じる:

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

= - \frac{1 \cdot 6}{12}

数を乗じる:

1 \cdot 6 = 6

= - \frac{6}{12}

共通因数を約分する:

6

= - \frac{1}{2}

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

指数の規則を適用する:

n

が偶数であれば

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

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

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

指数の規則を適用する:

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

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

規則を適用

1^{a} = 1

1^{2} = 1

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

2^{2} = 4

= \frac{1}{4}

= 1 - \frac{1}{4}

結合

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

1 - \frac{1}{4}

元を分数に変換する:

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

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

分母が等しいので, 分数を組み合わせる:

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

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

1 \cdot 4 - 1 = 3

1 \cdot 4 - 1

数を乗じる:

1 \cdot 4 = 4

= 4 - 1

数を引く:

4 - 1 = 3

= 3

= \frac{3}{4}

= \frac{3}{4}

累乗根の規則を適用する:

, 以下を想定

a \geq 0, b \geq 0

= \frac{3}{4}

4 = 2

4

数を因数に分解する:

4 = 2^{2}

= 2^{2}

累乗根の規則を適用する:

2^{2} = 2

= 2

= \frac{3}{2}

= 63 \frac{1}{12} \cdot \frac{3}{2}

分数を乗じる:

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

= \frac{1 \cdot 3 \cdot 6 3}{12 \cdot 2}

1 \cdot 3 \cdot 63 = 18

1 \cdot 3 \cdot 63

数を乗じる:

1 \cdot 6 = 6

= 633

累乗根の規則を適用する:

a a = a

33 = 3

= 6 \cdot 3

数を乗じる:

6 \cdot 3 = 18

= 18

= \frac{18}{12 \cdot 2}

数を乗じる:

12 \cdot 2 = 24

= \frac{18}{24}

共通因数を約分する:

6

= \frac{3}{4}

= - \frac{1}{4} - \frac{3}{4}

規則を適用

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

= \frac{- 1 - 3}{4}

数を引く:

- 1 - 3 = - 4

= \frac{- 4}{4}

分数の規則を適用する:

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

= - \frac{4}{4}

規則を適用

\frac{a}{a} = 1

= - 1

- 1 = - 1

真

解は

x = - \frac{1}{12}

x = - \frac{1}{12}

元のequationに当てはめて解を検算する

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

に当てはめて解を確認する

equationに一致しないものを削除する。

解答を確認する

- \frac{1}{12} :

真

- \frac{1}{12}

挿入

n = 1

- \frac{1}{12}

arcsin (6 x) + arcsin (63 x) = - \frac{π}{2}

の挿入向け

x = - \frac{1}{12}

arcsin (6 (- \frac{1}{12})) + arcsin (63 (- \frac{1}{12})) = - \frac{π}{2}

改良

- 1.57079 \dots = - 1.57079 \dots

\Rightarrow 真

x = - \frac{1}{12}

グラフ

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