What is (2sqrt(3)+2i)^8 ?

The solution to (2sqrt(3)+2i)^8 is -32768-32768sqrt(3)i

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simplify (2sqrt(3)+2i)^8

simplify

(23 + 2 i)^{8}

- 32768 - 327683 i

Solution steps

(23 + 2 i)^{8}

Apply binomial theorem:

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

a = 23, b = 2 i

= i = 0 \sum 8 (i 8) (23)^{(8 - i)} (2 i)^{i}

Expand summation

= \frac{8 !}{0 ! ( 8 - 0 ) !} (23)^{8} (2 i)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} (23)^{7} (2 i)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} (23)^{6} (2 i)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} (23)^{5} (2 i)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} (23)^{4} (2 i)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} (23)^{3} (2 i)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} (23)^{2} (2 i)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} (23)^{1} (2 i)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} (23)^{0} (2 i)^{8}

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} (23)^{8} (2 i)^{0} : 20736

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} (23)^{7} (2 i)^{1} : 552963 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} (23)^{6} (2 i)^{2} : 193536 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} (23)^{5} (2 i)^{3} : 1290243 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} (23)^{4} (2 i)^{4} : 161280 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} (23)^{3} (2 i)^{5} : 430083 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} (23)^{2} (2 i)^{6} : 21504 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} (23)^{1} (2 i)^{7} : 20483 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} (23)^{0} (2 i)^{8} : 256 i^{8}

= 20736 + 552963 i + 193536 i^{2} + 1290243 i^{3} + 161280 i^{4} + 430083 i^{5} + 21504 i^{6} + 20483 i^{7} + 256 i^{8}

Simplify

20736 + 552963 i + 193536 i^{2} + 1290243 i^{3} + 161280 i^{4} + 430083 i^{5} + 21504 i^{6} + 20483 i^{7} + 256 i^{8} : - 327683 i - 32768

= - 327683 i - 32768

Rewrite in standard complex form:

- 32768 - 327683 i

= - 32768 - 327683 i

(2 x + y) d x + (4 x + 2 y - 1) d y

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(2 x + 1) (2 x - 1) (x - 2 - 5) (x - 2 + 5)

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

(\frac{3 x y ^{2}}{8} - \frac{7 y}{2})^{3}

What is (2sqrt(3)+2i)^8 ?
The solution to (2sqrt(3)+2i)^8 is -32768-32768sqrt(3)i