simplify 6(x+iy)^5

Solution

simplify

6 (x + i y)^{5}

(6 x^{5} - 60 x^{3} y^{2} + 30 x y^{4}) + (30 x^{4} y - 60 x^{2} y^{3} + 6 y^{5}) i

Solution steps

6 (x + i y)^{5}

Expand

(x + i 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}

= 6 (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})

Distribute parentheses

= 6 x^{5} + 6 \cdot 5 i x^{4} y + 6 \cdot 10 i^{2} x^{3} y^{2} + 6 \cdot 10 i^{3} x^{2} y^{3} + 6 \cdot 5 i^{4} x y^{4} + 6 i^{5} y^{5}

Simplify

6 x^{5} + 6 \cdot 5 i x^{4} y + 6 \cdot 10 i^{2} x^{3} y^{2} + 6 \cdot 10 i^{3} x^{2} y^{3} + 6 \cdot 5 i^{4} x y^{4} + 6 i^{5} y^{5} : 6 x^{5} + 30 x^{4} y i - 60 x^{3} y^{2} - 60 x^{2} i y^{3} + 30 x y^{4} + 6 i y^{5}

= 6 x^{5} + 30 x^{4} y i - 60 x^{3} y^{2} - 60 x^{2} i y^{3} + 30 x y^{4} + 6 i y^{5}

Rewrite in standard complex form:

(6 x^{5} - 60 x^{3} y^{2} + 30 x y^{4}) + (30 x^{4} y - 60 x^{2} y^{3} + 6 y^{5}) i

= (6 x^{5} - 60 x^{3} y^{2} + 30 x y^{4}) + (30 x^{4} y - 60 x^{2} y^{3} + 6 y^{5}) i

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

\frac{1}{2} (- \frac{ln ( 5 x )}{2 x ^{2}} - \frac{1}{4 x ^{2}}) + C

t (x) x^{^{''}} - (t (x) + 3) x^{^{'}} + 3 x

(2 x + 3) x^{2} + 3 x

\frac{( n ( n + 1 ) ( 2 n + 1 ) )}{6}

What is 6(x+iy)^5 ?
The solution to 6(x+iy)^5 is (6x^5-60x^3y^2+30xy^4)+(30x^4y-60x^2y^3+6y^5)i