簡約 (r+3s)/(s+r)-(3s^2)/(s^2-r^2)+r/(s-r)

解

簡約

\frac{r + 3 s}{s + r} - \frac{3 s ^{2}}{s ^{2} - r ^{2}} + \frac{r}{s - r}

- \frac{rs}{s ^{2} - r ^{2}}

解答ステップ

\frac{r + 3 s}{s + r} - \frac{3 s ^{2}}{s ^{2} - r ^{2}} + \frac{r}{s - r}

以下の最小公倍数:

s + r, s^{2} - r^{2}, s - r : (r + s) (s - r) (s^{2} - r^{2})

LCMに基づいて分数を調整する

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

分数の規則を適用する:

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

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

簡素化

(r + 3 s) (s - r) (s^{2} - r^{2}) - 3 s^{2} (r + s) (s - r) + r (r + s) (s^{2} - r^{2}) : - r s^{3} + r^{3} s

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

キャンセル

\frac{- r s ^{3} + r ^{3} s}{( r + s ) ( s - r ) ( s ^{2} - r ^{2} )} : - \frac{rs}{s ^{2} - r ^{2}}

= - \frac{rs}{s ^{2} - r ^{2}}