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受欢迎的代数 >

化简 (x+iy)^6

解答

化简

(x + i y)^{6}

解答

(x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6}) + (6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5}) i

求解步骤

(x + i y)^{6}

使用二项式定理:

(a + b)^{n} = i = 0 \sum n (i n) a^{(n - i)} b^{i}

a = x, b = i y

= i = 0 \sum 6 (i 6) x^{(6 - i)} (i y)^{i}

展开求和

= \frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} (i y)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} (i y)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} (i y)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} (i y)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} (i y)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} (i y)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} (i y)^{6}

化简

\frac{6 !}{0 ! ( 6 - 0 ) !} x^{6} (i y)^{0} : x^{6}

化简

\frac{6 !}{1 ! ( 6 - 1 ) !} x^{5} (i y)^{1} : 6 i x^{5} y

化简

\frac{6 !}{2 ! ( 6 - 2 ) !} x^{4} (i y)^{2} : 15 i^{2} x^{4} y^{2}

化简

\frac{6 !}{3 ! ( 6 - 3 ) !} x^{3} (i y)^{3} : 20 i^{3} x^{3} y^{3}

化简

\frac{6 !}{4 ! ( 6 - 4 ) !} x^{2} (i y)^{4} : 15 i^{4} x^{2} y^{4}

化简

\frac{6 !}{5 ! ( 6 - 5 ) !} x^{1} (i y)^{5} : 6 i^{5} x y^{5}

化简

\frac{6 !}{6 ! ( 6 - 6 ) !} x^{0} (i y)^{6} : i^{6} y^{6}

= x^{6} + 6 i x^{5} y + 15 i^{2} x^{4} y^{2} + 20 i^{3} x^{3} y^{3} + 15 i^{4} x^{2} y^{4} + 6 i^{5} x y^{5} + i^{6} y^{6}

15 i^{2} x^{4} y^{2} = - 15 x^{4} y^{2}

20 i^{3} x^{3} y^{3} = - 20 i x^{3} y^{3}

15 i^{4} x^{2} y^{4} = 15 x^{2} y^{4}

i^{5} = i

i^{6} y^{6} = - y^{6}

= x^{6} + 6 i x^{5} y - 15 x^{4} y^{2} - 20 i x^{3} y^{3} + 15 x^{2} y^{4} + 6 i x y^{5} - y^{6}

Rewrite in standard complex form:

(x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6}) + (6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5}) i

= (x^{6} - 15 x^{4} y^{2} + 15 x^{2} y^{4} - y^{6}) + (6 x^{5} y - 20 x^{3} y^{3} + 6 x y^{5}) i

流行的例子

化简 6(-3w+1/3)

s im pl i f y 6 (- 3 w + \frac{1}{3})

化简 (sqrt(x)+1)(sqrt(x)-1)

s im pl i f y (x + 1) (x - 1)

展开-2/((x-1)^2)

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

展开 ln(sqrt(x/y))

e x p an d ln (\frac{x}{y})

展开 (x^2+y^2)dx

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