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

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

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

simplify

(3 - 33 i)^{5}

3888 + 38883 i

Solution steps

(3 - 33 i)^{5}

Apply binomial theorem:

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

a = 3, b = - 33 i

= i = 0 \sum 5 (i 5) \cdot 3^{(5 - i)} (- 33 i)^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 3^{5} (- 33 i)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 3^{4} (- 33 i)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 3^{3} (- 33 i)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 3^{2} (- 33 i)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 3^{1} (- 33 i)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 3^{0} (- 33 i)^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 3^{5} (- 33 i)^{0} : 243

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 3^{4} (- 33 i)^{1} : - 12153 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 3^{3} (- 33 i)^{2} : 7290 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 3^{2} (- 33 i)^{3} : - 72903 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 3^{1} (- 33 i)^{4} : 10935 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 3^{0} (- 33 i)^{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

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