What is expand (x+yi)^5 ?

The solution to expand (x+yi)^5 is (x^5-10x^3y^2+5xy^4)+(5x^4y-10x^2y^3+y^5)i

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expand (x+yi)^5

expand

(x + y i)^{5}

(x^{5} - 10 x^{3} y^{2} + 5 x y^{4}) + (5 x^{4} y - 10 x^{2} y^{3} + y^{5}) i

Solution steps

(x + y i)^{5}

Apply binomial theorem:

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

a = x, b = y i

= i = 0 \sum 5 (i 5) x^{(5 - i)} (y i)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} x^{5} (y i)^{0} : x^{5}

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} x^{4} (y i)^{1} : 5 i x^{4} y

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} x^{3} (y i)^{2} : 10 i^{2} x^{3} y^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} x^{2} (y i)^{3} : 10 i^{3} x^{2} y^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} x^{1} (y i)^{4} : 5 i^{4} x y^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} x^{0} (y i)^{5} : i^{5} y^{5}

= x^{5} + 5 i x^{4} y + 10 i^{2} x^{3} y^{2} + 10 i^{3} x^{2} y^{3} + 5 i^{4} x y^{4} + i^{5} y^{5}

10 i^{2} x^{3} y^{2} = - 10 x^{3} y^{2}

10 i^{3} x^{2} y^{3} = - 10 i x^{2} y^{3}

5 i^{4} x y^{4} = 5 x y^{4}

i^{5} = i

= x^{5} + 5 i x^{4} y - 10 x^{3} y^{2} - 10 i x^{2} y^{3} + 5 x y^{4} + i y^{5}

Rewrite in standard complex form:

(x^{5} - 10 x^{3} y^{2} + 5 x y^{4}) + (5 x^{4} y - 10 x^{2} y^{3} + y^{5}) i

= (x^{5} - 10 x^{3} y^{2} + 5 x y^{4}) + (5 x^{4} y - 10 x^{2} y^{3} + y^{5}) i

\frac{( x ^{2} + 1 + x ^{2} - 1 ) ^{2}}{2}

(x^{2} - \frac{5}{4}) e^{x}

\frac{5 x}{( 1 - x ) ^{2}}

A (- 4, - 5)

\frac{d}{d t} ((x) (- 6 x + 2 y))

What is expand (x+yi)^5 ?
The solution to expand (x+yi)^5 is (x^5-10x^3y^2+5xy^4)+(5x^4y-10x^2y^3+y^5)i