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

The solution to (-sqrt(3)-i)^5 is 16sqrt(3)-16i

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

simplify

(- 3 - i)^{5}

163 - 16 i

Solution steps

(- 3 - i)^{5}

Apply binomial theorem:

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

a = - 3, b = - i

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

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (- 3)^{5} (- i)^{0} : - 93

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (- 3)^{4} (- i)^{1} : - 45 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (- 3)^{3} (- i)^{2} : - 303 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (- 3)^{2} (- i)^{3} : - 30 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (- 3)^{1} (- i)^{4} : - 53 i^{4}

Simplify

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

= - 93 - 45 i - 303 i^{2} - 30 i^{3} - 53 i^{4} - i^{5}

Simplify

- 93 - 45 i - 303 i^{2} - 30 i^{3} - 53 i^{4} - i^{5} : - 16 i + 163

= - 16 i + 163

Rewrite in standard complex form:

163 - 16 i

= 163 - 16 i

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