展开 ((n(n+1))/2)^2

解答

展开

(\frac{n ( n + 1 )}{2})^{2}

\frac{n ^{4}}{4} + \frac{n ^{3}}{2} + \frac{n ^{2}}{4}

求解步骤

(\frac{n ( n + 1 )}{2})^{2}

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

= (\frac{n ^{2} + n}{2})^{2}

使用指数法则:

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

= \frac{( n ^{2} + n ) ^{2}}{2 ^{2}}

(n^{2} + n)^{2} = n^{4} + 2 n^{3} + n^{2}

= \frac{n ^{4} + 2 n ^{3} + n ^{2}}{2 ^{2}}

2^{2} = 4

= \frac{n ^{4} + 2 n ^{3} + n ^{2}}{4}

使用分式法则:

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

\frac{n ^{4} + 2 n ^{3} + n ^{2}}{4} = \frac{n ^{4}}{4} + \frac{2 n ^{3}}{4} + \frac{n ^{2}}{4}

= \frac{n ^{4}}{4} + \frac{2 n ^{3}}{4} + \frac{n ^{2}}{4}

消掉

\frac{2 n ^{3}}{4} : \frac{n ^{3}}{2}

= \frac{n ^{4}}{4} + \frac{n ^{3}}{2} + \frac{n ^{2}}{4}

s im pl i f y (1 + i)^{25}

s im pl i f y (3 + x)^{2}

e x p an d (x + \frac{1}{2})^{2}

e x p an d (x^{\frac{1}{2}} + y^{\frac{1}{2}}) (x^{\frac{1}{2}} - y^{\frac{1}{2}})

e x p an d \frac{x}{( 1 - x ) ^{2}}