The solution to (x+yi)^6 is (x^6-15x^4y^2+15x^2y^4-y^6)+(6x^5y-20x^3y^3+6xy^5)i

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simplify (x+yi)^6

Solution

simplify

(x + y i)^{6}

(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

Solution steps

(x + y i)^{6}

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 6 (i 6) x^{(6 - i)} (y i)^{i}

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

e^{3 x} (10 - 6 y + e^{- 3 x})

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

\frac{20 n ( n + 1 )}{2}

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

(x + \frac{1}{x})^{9}

What is (x+yi)^6 ?
The solution to (x+yi)^6 is (x^6-15x^4y^2+15x^2y^4-y^6)+(6x^5y-20x^3y^3+6xy^5)i