What is (3-sqrt(3)i)^7 ?

The solution to (3-sqrt(3)i)^7 is -5184+1728sqrt(3)i

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simplify (3-sqrt(3)i)^7

simplify

(3 - 3 i)^{7}

- 5184 + 17283 i

Solution steps

(3 - 3 i)^{7}

Apply binomial theorem:

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

a = 3, b = - 3 i

= i = 0 \sum 7 (i 7) \cdot 3^{(7 - i)} (- 3 i)^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} \cdot 3^{7} (- 3 i)^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} \cdot 3^{6} (- 3 i)^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} \cdot 3^{5} (- 3 i)^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} \cdot 3^{4} (- 3 i)^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} \cdot 3^{3} (- 3 i)^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} \cdot 3^{2} (- 3 i)^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} \cdot 3^{1} (- 3 i)^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} \cdot 3^{0} (- 3 i)^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} \cdot 3^{7} (- 3 i)^{0} : 2187

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} \cdot 3^{6} (- 3 i)^{1} : - 51033 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} \cdot 3^{5} (- 3 i)^{2} : 15309 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} \cdot 3^{4} (- 3 i)^{3} : - 85053 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} \cdot 3^{3} (- 3 i)^{4} : 8505 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} \cdot 3^{2} (- 3 i)^{5} : - 17013 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} \cdot 3^{1} (- 3 i)^{6} : 567 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} \cdot 3^{0} (- 3 i)^{7} : - 273 i^{7}

= 2187 - 51033 i + 15309 i^{2} - 85053 i^{3} + 8505 i^{4} - 17013 i^{5} + 567 i^{6} - 273 i^{7}

Simplify

2187 - 51033 i + 15309 i^{2} - 85053 i^{3} + 8505 i^{4} - 17013 i^{5} + 567 i^{6} - 273 i^{7} : 17283 i - 5184

= 17283 i - 5184

Rewrite in standard complex form:

- 5184 + 17283 i

= - 5184 + 17283 i

(e - f k) (e - e f k + f k)

\frac{248 ( s + 5 )}{s ( s + 4 ) ( s ^{2} + 4 )}

(x^{2} - 2 x + 9)^{7}

(v + v^{2} + 1)^{2}

\frac{2 x ^{2} + 9 x - 5}{6 x ( 4 x - 5 )}