The solution to (a+bi)^6 is (a^6-15a^4b^2+15a^2b^4-b^6)+(6a^5b-20a^3b^3+6ab^5)i

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simplify (a+bi)^6

Solution

simplify

(a + bi)^{6}

(a^{6} - 15 a^{4} b^{2} + 15 a^{2} b^{4} - b^{6}) + (6 a^{5} b - 20 a^{3} b^{3} + 6 a b^{5}) i

Solution steps

(a + bi)^{6}

Apply binomial theorem:

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

a = a, b = bi

= i = 0 \sum 6 (i 6) a^{(6 - i)} (bi)^{i}

Expand summation

= \frac{6 !}{0 ! ( 6 - 0 ) !} a^{6} (bi)^{0} + \frac{6 !}{1 ! ( 6 - 1 ) !} a^{5} (bi)^{1} + \frac{6 !}{2 ! ( 6 - 2 ) !} a^{4} (bi)^{2} + \frac{6 !}{3 ! ( 6 - 3 ) !} a^{3} (bi)^{3} + \frac{6 !}{4 ! ( 6 - 4 ) !} a^{2} (bi)^{4} + \frac{6 !}{5 ! ( 6 - 5 ) !} a^{1} (bi)^{5} + \frac{6 !}{6 ! ( 6 - 6 ) !} a^{0} (bi)^{6}

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} a^{6} (bi)^{0} : a^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} a^{5} (bi)^{1} : 6 i a^{5} b

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} a^{4} (bi)^{2} : 15 i^{2} a^{4} b^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} a^{3} (bi)^{3} : 20 i^{3} a^{3} b^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} a^{2} (bi)^{4} : 15 i^{4} a^{2} b^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} a^{1} (bi)^{5} : 6 i^{5} a b^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} a^{0} (bi)^{6} : i^{6} b^{6}

= a^{6} + 6 i a^{5} b + 15 i^{2} a^{4} b^{2} + 20 i^{3} a^{3} b^{3} + 15 i^{4} a^{2} b^{4} + 6 i^{5} a b^{5} + i^{6} b^{6}

15 i^{2} a^{4} b^{2} = - 15 a^{4} b^{2}

20 i^{3} a^{3} b^{3} = - 20 i a^{3} b^{3}

15 i^{4} a^{2} b^{4} = 15 a^{2} b^{4}

i^{5} = i

i^{6} b^{6} = - b^{6}

= a^{6} + 6 i a^{5} b - 15 a^{4} b^{2} - 20 i a^{3} b^{3} + 15 a^{2} b^{4} + 6 ia b^{5} - b^{6}

Rewrite in standard complex form:

(a^{6} - 15 a^{4} b^{2} + 15 a^{2} b^{4} - b^{6}) + (6 a^{5} b - 20 a^{3} b^{3} + 6 a b^{5}) i

= (a^{6} - 15 a^{4} b^{2} + 15 a^{2} b^{4} - b^{6}) + (6 a^{5} b - 20 a^{3} b^{3} + 6 a b^{5}) i

(1 + cos (t)) (e^{t^{2}} + t)^{2}

(x^{2})^{2}

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

sin (x) (sin (x) + cos (x))

(- 4, - 3) y (8, 0)

What is (a+bi)^6 ?
The solution to (a+bi)^6 is (a^6-15a^4b^2+15a^2b^4-b^6)+(6a^5b-20a^3b^3+6ab^5)i