The solution to (x+iy)^8 is (x^8-28x^6y^2+70x^4y^4-28x^2y^6+y^8)+(8x^7y-56x^5y^3+56x^3y^5-8xy^7)i

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simplify (x+iy)^8

Solution

simplify

(x + i y)^{8}

(x^{8} - 28 x^{6} y^{2} + 70 x^{4} y^{4} - 28 x^{2} y^{6} + y^{8}) + (8 x^{7} y - 56 x^{5} y^{3} + 56 x^{3} y^{5} - 8 x y^{7}) i

Solution steps

(x + i y)^{8}

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 8 (i 8) x^{(8 - i)} (i y)^{i}

Expand summation

= \frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} (i y)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} (i y)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} (i y)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} (i y)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} (i y)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} (i y)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} (i y)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} (i y)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} (i y)^{8}

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} x^{8} (i y)^{0} : x^{8}

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} x^{7} (i y)^{1} : 8 i x^{7} y

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} x^{6} (i y)^{2} : 28 i^{2} x^{6} y^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} x^{5} (i y)^{3} : 56 i^{3} x^{5} y^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} x^{4} (i y)^{4} : 70 i^{4} x^{4} y^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} x^{3} (i y)^{5} : 56 i^{5} x^{3} y^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} x^{2} (i y)^{6} : 28 i^{6} x^{2} y^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} x^{1} (i y)^{7} : 8 i^{7} x y^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} x^{0} (i y)^{8} : i^{8} y^{8}

= x^{8} + 8 i x^{7} y + 28 i^{2} x^{6} y^{2} + 56 i^{3} x^{5} y^{3} + 70 i^{4} x^{4} y^{4} + 56 i^{5} x^{3} y^{5} + 28 i^{6} x^{2} y^{6} + 8 i^{7} x y^{7} + i^{8} y^{8}

28 i^{2} x^{6} y^{2} = - 28 x^{6} y^{2}

56 i^{3} x^{5} y^{3} = - 56 i x^{5} y^{3}

70 i^{4} x^{4} y^{4} = 70 x^{4} y^{4}

i^{5} = i

28 i^{6} x^{2} y^{6} = - 28 x^{2} y^{6}

8 i^{7} x y^{7} = - 8 i x y^{7}

i^{8} y^{8} = y^{8}

= x^{8} + 8 i x^{7} y - 28 x^{6} y^{2} - 56 i x^{5} y^{3} + 70 x^{4} y^{4} + 56 i x^{3} y^{5} - 28 x^{2} y^{6} - 8 i x y^{7} + y^{8}

Rewrite in standard complex form:

(x^{8} - 28 x^{6} y^{2} + 70 x^{4} y^{4} - 28 x^{2} y^{6} + y^{8}) + (8 x^{7} y - 56 x^{5} y^{3} + 56 x^{3} y^{5} - 8 x y^{7}) i

= (x^{8} - 28 x^{6} y^{2} + 70 x^{4} y^{4} - 28 x^{2} y^{6} + y^{8}) + (8 x^{7} y - 56 x^{5} y^{3} + 56 x^{3} y^{5} - 8 x y^{7}) i

s im pl i f y (x + i y)^{7}

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

e x p an d (1 + x y) e^{x y}

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

e x p an d - \frac{12}{s ( s ^{2} - 3 s - 10 )}

What is (x+iy)^8 ?
The solution to (x+iy)^8 is (x^8-28x^6y^2+70x^4y^4-28x^2y^6+y^8)+(8x^7y-56x^5y^3+56x^3y^5-8xy^7)i