展开 ((2+x)^5)/(32)

解答

展开

\frac{( 2 + x ) ^{5}}{32}

\frac{x ^{5}}{32} + \frac{5 x ^{4}}{16} + \frac{5 x ^{3}}{4} + \frac{5 x ^{2}}{2} + \frac{5 x}{2} + 1

求解步骤

\frac{( 2 + x ) ^{5}}{32}

(2 + x)^{5} = 32 + 80 x + 80 x^{2} + 40 x^{3} + 10 x^{4} + x^{5}

= \frac{32 + 80 x + 80 x ^{2} + 40 x ^{3} + 10 x ^{4} + x ^{5}}{32}

使用分式法则:

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

\frac{32 + 80 x + 80 x ^{2} + 40 x ^{3} + 10 x ^{4} + x ^{5}}{32} = \frac{32}{32} + \frac{80 x}{32} + \frac{80 x ^{2}}{32} + \frac{40 x ^{3}}{32} + \frac{10 x ^{4}}{32} + \frac{x ^{5}}{32}

= \frac{32}{32} + \frac{80 x}{32} + \frac{80 x ^{2}}{32} + \frac{40 x ^{3}}{32} + \frac{10 x ^{4}}{32} + \frac{x ^{5}}{32}

使用法则

\frac{a}{a} = 1

\frac{32}{32} = 1

= 1 + \frac{80 x}{32} + \frac{80 x ^{2}}{32} + \frac{40 x ^{3}}{32} + \frac{10 x ^{4}}{32} + \frac{x ^{5}}{32}

\frac{80 x}{32} = \frac{5 x}{2}

\frac{80 x ^{2}}{32} = \frac{5 x ^{2}}{2}

\frac{40 x ^{3}}{32} = \frac{5 x ^{3}}{4}

\frac{10 x ^{4}}{32} = \frac{5 x ^{4}}{16}

= 1 + \frac{5 x}{2} + \frac{5 x ^{2}}{2} + \frac{5 x ^{3}}{4} + \frac{5 x ^{4}}{16} + \frac{x ^{5}}{32}

改写为标准形式

= \frac{x ^{5}}{32} + \frac{5 x ^{4}}{16} + \frac{5 x ^{3}}{4} + \frac{5 x ^{2}}{2} + \frac{5 x}{2} + 1

e x p an d \frac{3 ( 40 x ^{3} + 5 ) ( 10 x ^{4} + 5 x + 3 ) ^{\frac{1}{2}}}{2}

e x p an d 2 (ln (2 x) + β) + 3 e^{β} x

e x p an d (4 x^{3} y^{2} + 5 x^{2} y^{3}) d x + (5 x^{3} y^{2} - 4 x^{2} y^{3}) d y

e x p an d (x y^{2} + 4 x^{2} y) d x + (x + y) x^{2} d y

e x p an d L (x) (x^{'} + 3 x)