簡約 (a-b)/(3a+3b)-(a-2b)/(6a-6b)+(a^2+2ab-6b^2)/(9a^2-9b^2)

解

簡約

\frac{a - b}{3 a + 3 b} - \frac{a - 2 b}{6 a - 6 b} + \frac{a ^{2} + 2 ab - 6 b ^{2}}{9 a ^{2} - 9 b ^{2}}

\frac{5 a}{18 ( a + b )}

解答ステップ

\frac{a - b}{3 a + 3 b} - \frac{a - 2 b}{6 a - 6 b} + \frac{a ^{2} + 2 ab - 6 b ^{2}}{9 a ^{2} - 9 b ^{2}}

以下の最小公倍数:

3 a + 3 b, 6 a - 6 b, 9 a^{2} - 9 b^{2} : (3 a + 3 b) (6 a - 6 b) (9 a^{2} - 9 b^{2})

LCMに基づいて分数を調整する

= \frac{( a - b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )} - \frac{( a - 2 b ) ( 3 a + 3 b ) ( 9 a ^{2} - 9 b ^{2} )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )} + \frac{( a ^{2} + 2 ab - 6 b ^{2} ) ( 3 a + 3 b ) ( 6 a - 6 b )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}

分数の規則を適用する:

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

= \frac{( a - b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} ) - ( a - 2 b ) ( 3 a + 3 b ) ( 9 a ^{2} - 9 b ^{2} ) + ( a ^{2} + 2 ab - 6 b ^{2} ) ( 3 a + 3 b ) ( 6 a - 6 b )}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}

簡素化

(a - b) (6 a - 6 b) (9 a^{2} - 9 b^{2}) - (a - 2 b) (3 a + 3 b) (9 a^{2} - 9 b^{2}) + (a^{2} + 2 ab - 6 b^{2}) (3 a + 3 b) (6 a - 6 b) : 45 a^{4} - 45 a^{2} b^{2} - 45 a^{3} b + 45 a b^{3}

= \frac{45 a ^{4} - 45 a ^{2} b ^{2} - 45 a ^{3} b + 45 a b ^{3}}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )}

キャンセル

\frac{45 a ^{4} - 45 a ^{2} b ^{2} - 45 a ^{3} b + 45 a b ^{3}}{( 3 a + 3 b ) ( 6 a - 6 b ) ( 9 a ^{2} - 9 b ^{2} )} : \frac{5 a}{18 ( a + b )}

= \frac{5 a}{18 ( a + b )}