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

\frac{x + 1}{x ^{\frac{2}{3}} ( x + 3 ) ^{\frac{1}{3}}}

\frac{2 ( x ^{4} - 1 )}{x ^{3}}

(2 - e^{- x})^{2}

\frac{3}{( 1 - x ) ^{2}}

2 (62 + 19) (62 - 19)

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