What is (1-sqrt(3)i)^{16} ?

The solution to (1-sqrt(3)i)^{16} is -32768+32768sqrt(3)i

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simplify (1-sqrt(3)i)^{16}

Solution

simplify

(1 - 3 i)^{16}

Solution

- 32768 + 327683 i

Solution steps

(1 - 3 i)^{16}

Apply binomial theorem:

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

a = 1, b = - 3 i

= i = 0 \sum 16 (i 16) \cdot 1^{(16 - i)} (- 3 i)^{i}

Expand summation

= \frac{16 !}{0 ! ( 16 - 0 ) !} \cdot 1^{16} (- 3 i)^{0} + \frac{16 !}{1 ! ( 16 - 1 ) !} \cdot 1^{15} (- 3 i)^{1} + \frac{16 !}{2 ! ( 16 - 2 ) !} \cdot 1^{14} (- 3 i)^{2} + \frac{16 !}{3 ! ( 16 - 3 ) !} \cdot 1^{13} (- 3 i)^{3} + \frac{16 !}{4 ! ( 16 - 4 ) !} \cdot 1^{12} (- 3 i)^{4} + \frac{16 !}{5 ! ( 16 - 5 ) !} \cdot 1^{11} (- 3 i)^{5} + \frac{16 !}{6 ! ( 16 - 6 ) !} \cdot 1^{10} (- 3 i)^{6} + \frac{16 !}{7 ! ( 16 - 7 ) !} \cdot 1^{9} (- 3 i)^{7} + \frac{16 !}{8 ! ( 16 - 8 ) !} \cdot 1^{8} (- 3 i)^{8} + \frac{16 !}{9 ! ( 16 - 9 ) !} \cdot 1^{7} (- 3 i)^{9} + \frac{16 !}{10 ! ( 16 - 10 ) !} \cdot 1^{6} (- 3 i)^{10} + \frac{16 !}{11 ! ( 16 - 11 ) !} \cdot 1^{5} (- 3 i)^{11} + \frac{16 !}{12 ! ( 16 - 12 ) !} \cdot 1^{4} (- 3 i)^{12} + \frac{16 !}{13 ! ( 16 - 13 ) !} \cdot 1^{3} (- 3 i)^{13} + \frac{16 !}{14 ! ( 16 - 14 ) !} \cdot 1^{2} (- 3 i)^{14} + \frac{16 !}{15 ! ( 16 - 15 ) !} \cdot 1^{1} (- 3 i)^{15} + \frac{16 !}{16 ! ( 16 - 16 ) !} \cdot 1^{0} (- 3 i)^{16}

Simplify

\frac{16 !}{0 ! ( 16 - 0 ) !} \cdot 1^{16} (- 3 i)^{0} : 1

Simplify

\frac{16 !}{1 ! ( 16 - 1 ) !} \cdot 1^{15} (- 3 i)^{1} : - 163 i

Simplify

\frac{16 !}{2 ! ( 16 - 2 ) !} \cdot 1^{14} (- 3 i)^{2} : 360 i^{2}

Simplify

\frac{16 !}{3 ! ( 16 - 3 ) !} \cdot 1^{13} (- 3 i)^{3} : - 16803 i^{3}

Simplify

\frac{16 !}{4 ! ( 16 - 4 ) !} \cdot 1^{12} (- 3 i)^{4} : 16380 i^{4}

Simplify

\frac{16 !}{5 ! ( 16 - 5 ) !} \cdot 1^{11} (- 3 i)^{5} : - 393123 i^{5}

Simplify

\frac{16 !}{6 ! ( 16 - 6 ) !} \cdot 1^{10} (- 3 i)^{6} : 216216 i^{6}

Simplify

\frac{16 !}{7 ! ( 16 - 7 ) !} \cdot 1^{9} (- 3 i)^{7} : - 3088803 i^{7}

Simplify

\frac{16 !}{8 ! ( 16 - 8 ) !} \cdot 1^{8} (- 3 i)^{8} : 1042470 i^{8}

Simplify

\frac{16 !}{9 ! ( 16 - 9 ) !} \cdot 1^{7} (- 3 i)^{9} : - 9266403 i^{9}

Simplify

\frac{16 !}{10 ! ( 16 - 10 ) !} \cdot 1^{6} (- 3 i)^{10} : 1945944 i^{10}

Simplify

\frac{16 !}{11 ! ( 16 - 11 ) !} \cdot 1^{5} (- 3 i)^{11} : - 10614243 i^{11}

Simplify

\frac{16 !}{12 ! ( 16 - 12 ) !} \cdot 1^{4} (- 3 i)^{12} : 1326780 i^{12}

Simplify

\frac{16 !}{13 ! ( 16 - 13 ) !} \cdot 1^{3} (- 3 i)^{13} : - 4082403 i^{13}

Simplify

\frac{16 !}{14 ! ( 16 - 14 ) !} \cdot 1^{2} (- 3 i)^{14} : 262440 i^{14}

Simplify

\frac{16 !}{15 ! ( 16 - 15 ) !} \cdot 1^{1} (- 3 i)^{15} : - 349923 i^{15}

Simplify

\frac{16 !}{16 ! ( 16 - 16 ) !} \cdot 1^{0} (- 3 i)^{16} : 6561 i^{16}

= 1 - 163 i + 360 i^{2} - 16803 i^{3} + 16380 i^{4} - 393123 i^{5} + 216216 i^{6} - 3088803 i^{7} + 1042470 i^{8} - 9266403 i^{9} + 1945944 i^{10} - 10614243 i^{11} + 1326780 i^{12} - 4082403 i^{13} + 262440 i^{14} - 349923 i^{15} + 6561 i^{16}

Simplify

1 - 163 i + 360 i^{2} - 16803 i^{3} + 16380 i^{4} - 393123 i^{5} + 216216 i^{6} - 3088803 i^{7} + 1042470 i^{8} - 9266403 i^{9} + 1945944 i^{10} - 10614243 i^{11} + 1326780 i^{12} - 4082403 i^{13} + 262440 i^{14} - 349923 i^{15} + 6561 i^{16} : 327683 i - 32768

= 327683 i - 32768

Rewrite in standard complex form:

- 32768 + 327683 i

= - 32768 + 327683 i

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Frequently Asked Questions (FAQ)

What is (1-sqrt(3)i)^{16} ?
The solution to (1-sqrt(3)i)^{16} is -32768+32768sqrt(3)i