What is (-4sqrt(3)-4i)^5 ?

The solution to (-4sqrt(3)-4i)^5 is 16384sqrt(3)-16384i

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simplify (-4sqrt(3)-4i)^5

simplify

(- 43 - 4 i)^{5}

163843 - 16384 i

Solution steps

(- 43 - 4 i)^{5}

Apply binomial theorem:

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

a = - 43, b = - 4 i

= i = 0 \sum 5 (i 5) (- 43)^{(5 - i)} (- 4 i)^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} (- 43)^{5} (- 4 i)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} (- 43)^{4} (- 4 i)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} (- 43)^{3} (- 4 i)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} (- 43)^{2} (- 4 i)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} (- 43)^{1} (- 4 i)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} (- 43)^{0} (- 4 i)^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (- 43)^{5} (- 4 i)^{0} : - 92163

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (- 43)^{4} (- 4 i)^{1} : - 46080 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (- 43)^{3} (- 4 i)^{2} : - 307203 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (- 43)^{2} (- 4 i)^{3} : - 30720 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (- 43)^{1} (- 4 i)^{4} : - 51203 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (- 43)^{0} (- 4 i)^{5} : - 1024 i^{5}

= - 92163 - 46080 i - 307203 i^{2} - 30720 i^{3} - 51203 i^{4} - 1024 i^{5}

Simplify

- 92163 - 46080 i - 307203 i^{2} - 30720 i^{3} - 51203 i^{4} - 1024 i^{5} : - 16384 i + 163843

= - 16384 i + 163843

Rewrite in standard complex form:

163843 - 16384 i

= 163843 - 16384 i

\frac{t ( l ^{2} w ^{2} + r )}{r t + l ^{2} w ^{2} + r}

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

(x - 1 + i y)^{3}

(- 2 sin (t) - t cos (t))^{2}

(x - 1 + i y)^{2}

What is (-4sqrt(3)-4i)^5 ?
The solution to (-4sqrt(3)-4i)^5 is 16384sqrt(3)-16384i