What is (-3+3isqrt(3))^5 ?

The solution to (-3+3isqrt(3))^5 is -3888-3888sqrt(3)i

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simplify (-3+3isqrt(3))^5

simplify

(- 3 + 3 i 3)^{5}

- 3888 - 38883 i

Solution steps

(- 3 + 3 i 3)^{5}

Apply binomial theorem:

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

a = - 3, b = 3 i 3

= i = 0 \sum 5 (i 5) (- 3)^{(5 - i)} (3 i 3)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (- 3)^{5} (3 i 3)^{0} : - 243

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (- 3)^{4} (3 i 3)^{1} : 12153 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (- 3)^{3} (3 i 3)^{2} : - 7290 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (- 3)^{2} (3 i 3)^{3} : 72903 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (- 3)^{1} (3 i 3)^{4} : - 10935 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (- 3)^{0} (3 i 3)^{5} : 21873 i^{5}

= - 243 + 12153 i - 7290 i^{2} + 72903 i^{3} - 10935 i^{4} + 21873 i^{5}

Simplify

- 243 + 12153 i - 7290 i^{2} + 72903 i^{3} - 10935 i^{4} + 21873 i^{5} : - 38883 i - 3888

= - 38883 i - 3888

Rewrite in standard complex form:

- 3888 - 38883 i

= - 3888 - 38883 i

\frac{4 π ( c - a ) ( b - c ) EF}{E ( b - c ) + F ( c - a )}

3 (x - 13) (x + 13) (x - 1)^{2}

(x + 3) e^{2 x}

2 ((- 1)^{k - 1})

(x + 5 h) (e^{y + 5 h})

What is (-3+3isqrt(3))^5 ?
The solution to (-3+3isqrt(3))^5 is -3888-3888sqrt(3)i