The solution to (x+iy)^7 is (x^7-21x^5y^2+35x^3y^4-7xy^6)+(7x^6y-35x^4y^3+21x^2y^5-y^7)i

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simplify (x+iy)^7

Solution

simplify

(x + i y)^{7}

(x^{7} - 21 x^{5} y^{2} + 35 x^{3} y^{4} - 7 x y^{6}) + (7 x^{6} y - 35 x^{4} y^{3} + 21 x^{2} y^{5} - y^{7}) i

Solution steps

(x + i y)^{7}

Apply binomial theorem:

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

a = x, b = i y

= i = 0 \sum 7 (i 7) x^{(7 - i)} (i y)^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (i y)^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (i y)^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (i y)^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (i y)^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (i y)^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (i y)^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (i y)^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (i y)^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} x^{7} (i y)^{0} : x^{7}

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} x^{6} (i y)^{1} : 7 i x^{6} y

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} x^{5} (i y)^{2} : 21 i^{2} x^{5} y^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} x^{4} (i y)^{3} : 35 i^{3} x^{4} y^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} x^{3} (i y)^{4} : 35 i^{4} x^{3} y^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} x^{2} (i y)^{5} : 21 i^{5} x^{2} y^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} x^{1} (i y)^{6} : 7 i^{6} x y^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} x^{0} (i y)^{7} : i^{7} y^{7}

= x^{7} + 7 i x^{6} y + 21 i^{2} x^{5} y^{2} + 35 i^{3} x^{4} y^{3} + 35 i^{4} x^{3} y^{4} + 21 i^{5} x^{2} y^{5} + 7 i^{6} x y^{6} + i^{7} y^{7}

21 i^{2} x^{5} y^{2} = - 21 x^{5} y^{2}

35 i^{3} x^{4} y^{3} = - 35 i x^{4} y^{3}

35 i^{4} x^{3} y^{4} = 35 x^{3} y^{4}

i^{5} = i

7 i^{6} x y^{6} = - 7 x y^{6}

i^{7} = - i

= x^{7} + 7 i x^{6} y - 21 x^{5} y^{2} - 35 i x^{4} y^{3} + 35 x^{3} y^{4} + 21 i x^{2} y^{5} - 7 x y^{6} - i y^{7}

Rewrite in standard complex form:

(x^{7} - 21 x^{5} y^{2} + 35 x^{3} y^{4} - 7 x y^{6}) + (7 x^{6} y - 35 x^{4} y^{3} + 21 x^{2} y^{5} - y^{7}) i

= (x^{7} - 21 x^{5} y^{2} + 35 x^{3} y^{4} - 7 x y^{6}) + (7 x^{6} y - 35 x^{4} y^{3} + 21 x^{2} y^{5} - y^{7}) i

\frac{1}{( 1 - e ^{- π s} ) ( s ^{2} + 1 )}

(1 + x y) e^{x y}

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

- \frac{12}{s ( s ^{2} - 3 s - 10 )}

(y - 2) ln (y - 2)

What is (x+iy)^7 ?
The solution to (x+iy)^7 is (x^7-21x^5y^2+35x^3y^4-7xy^6)+(7x^6y-35x^4y^3+21x^2y^5-y^7)i