What is (-1+sqrt(3)i)^{25} ?

The solution to (-1+sqrt(3)i)^{25} is -16777216+16777216sqrt(3)i

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Popular >

simplify (-1+sqrt(3)i)^{25}

Solution

simplify

(- 1 + 3 i)^{25}

Solution

- 16777216 + 167772163 i

Solution steps

(- 1 + 3 i)^{25}

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 25 (i 25) (- 1)^{(25 - i)} (3 i)^{i}

Expand summation

= \frac{25 !}{0 ! ( 25 - 0 ) !} (- 1)^{25} (3 i)^{0} + \frac{25 !}{1 ! ( 25 - 1 ) !} (- 1)^{24} (3 i)^{1} + \frac{25 !}{2 ! ( 25 - 2 ) !} (- 1)^{23} (3 i)^{2} + \frac{25 !}{3 ! ( 25 - 3 ) !} (- 1)^{22} (3 i)^{3} + \frac{25 !}{4 ! ( 25 - 4 ) !} (- 1)^{21} (3 i)^{4} + \frac{25 !}{5 ! ( 25 - 5 ) !} (- 1)^{20} (3 i)^{5} + \frac{25 !}{6 ! ( 25 - 6 ) !} (- 1)^{19} (3 i)^{6} + \frac{25 !}{7 ! ( 25 - 7 ) !} (- 1)^{18} (3 i)^{7} + \frac{25 !}{8 ! ( 25 - 8 ) !} (- 1)^{17} (3 i)^{8} + \frac{25 !}{9 ! ( 25 - 9 ) !} (- 1)^{16} (3 i)^{9} + \frac{25 !}{10 ! ( 25 - 10 ) !} (- 1)^{15} (3 i)^{10} + \frac{25 !}{11 ! ( 25 - 11 ) !} (- 1)^{14} (3 i)^{11} + \frac{25 !}{12 ! ( 25 - 12 ) !} (- 1)^{13} (3 i)^{12} + \frac{25 !}{13 ! ( 25 - 13 ) !} (- 1)^{12} (3 i)^{13} + \frac{25 !}{14 ! ( 25 - 14 ) !} (- 1)^{11} (3 i)^{14} + \frac{25 !}{15 ! ( 25 - 15 ) !} (- 1)^{10} (3 i)^{15} + \frac{25 !}{16 ! ( 25 - 16 ) !} (- 1)^{9} (3 i)^{16} + \frac{25 !}{17 ! ( 25 - 17 ) !} (- 1)^{8} (3 i)^{17} + \frac{25 !}{18 ! ( 25 - 18 ) !} (- 1)^{7} (3 i)^{18} + \frac{25 !}{19 ! ( 25 - 19 ) !} (- 1)^{6} (3 i)^{19} + \frac{25 !}{20 ! ( 25 - 20 ) !} (- 1)^{5} (3 i)^{20} + \frac{25 !}{21 ! ( 25 - 21 ) !} (- 1)^{4} (3 i)^{21} + \frac{25 !}{22 ! ( 25 - 22 ) !} (- 1)^{3} (3 i)^{22} + \frac{25 !}{23 ! ( 25 - 23 ) !} (- 1)^{2} (3 i)^{23} + \frac{25 !}{24 ! ( 25 - 24 ) !} (- 1)^{1} (3 i)^{24} + \frac{25 !}{25 ! ( 25 - 25 ) !} (- 1)^{0} (3 i)^{25}

Simplify

\frac{25 !}{0 ! ( 25 - 0 ) !} (- 1)^{25} (3 i)^{0} : - 1

Simplify

\frac{25 !}{1 ! ( 25 - 1 ) !} (- 1)^{24} (3 i)^{1} : 253 i

Simplify

\frac{25 !}{2 ! ( 25 - 2 ) !} (- 1)^{23} (3 i)^{2} : - 900 i^{2}

Simplify

\frac{25 !}{3 ! ( 25 - 3 ) !} (- 1)^{22} (3 i)^{3} : 69003 i^{3}

Simplify

\frac{25 !}{4 ! ( 25 - 4 ) !} (- 1)^{21} (3 i)^{4} : - 113850 i^{4}

Simplify

\frac{25 !}{5 ! ( 25 - 5 ) !} (- 1)^{20} (3 i)^{5} : 4781703 i^{5}

Simplify

\frac{25 !}{6 ! ( 25 - 6 ) !} (- 1)^{19} (3 i)^{6} : - 4781700 i^{6}

Simplify

\frac{25 !}{7 ! ( 25 - 7 ) !} (- 1)^{18} (3 i)^{7} : 129789003 i^{7}

Simplify

\frac{25 !}{8 ! ( 25 - 8 ) !} (- 1)^{17} (3 i)^{8} : - 87607575 i^{8}

Simplify

\frac{25 !}{9 ! ( 25 - 9 ) !} (- 1)^{16} (3 i)^{9} : 1654809753 i^{9}

Simplify

\frac{25 !}{10 ! ( 25 - 10 ) !} (- 1)^{15} (3 i)^{10} : - 794308680 i^{10}

Simplify

\frac{25 !}{11 ! ( 25 - 11 ) !} (- 1)^{14} (3 i)^{11} : 10831482003 i^{11}

Simplify

\frac{25 !}{12 ! ( 25 - 12 ) !} (- 1)^{13} (3 i)^{12} : - 3791018700 i^{12}

Simplify

\frac{25 !}{13 ! ( 25 - 13 ) !} (- 1)^{12} (3 i)^{13} : 37910187003 i^{13}

Simplify

\frac{25 !}{14 ! ( 25 - 14 ) !} (- 1)^{11} (3 i)^{14} : - 9748333800 i^{14}

Simplify

\frac{25 !}{15 ! ( 25 - 15 ) !} (- 1)^{10} (3 i)^{15} : 71487781203 i^{15}

Simplify

\frac{25 !}{16 ! ( 25 - 16 ) !} (- 1)^{9} (3 i)^{16} : - 13403958975 i^{16}

Simplify

\frac{25 !}{17 ! ( 25 - 17 ) !} (- 1)^{8} (3 i)^{17} : 70962135753 i^{17}

Simplify

\frac{25 !}{18 ! ( 25 - 18 ) !} (- 1)^{7} (3 i)^{18} : - 9461618100 i^{18}

Simplify

\frac{25 !}{19 ! ( 25 - 19 ) !} (- 1)^{6} (3 i)^{19} : 34858593003 i^{19}

Simplify

\frac{25 !}{20 ! ( 25 - 20 ) !} (- 1)^{5} (3 i)^{20} : - 3137273370 i^{20}

Simplify

\frac{25 !}{21 ! ( 25 - 21 ) !} (- 1)^{4} (3 i)^{21} : 7469698503 i^{21}

Simplify

\frac{25 !}{22 ! ( 25 - 22 ) !} (- 1)^{3} (3 i)^{22} : - 407438100 i^{22}

Simplify

\frac{25 !}{23 ! ( 25 - 23 ) !} (- 1)^{2} (3 i)^{23} : 531441003 i^{23}

Simplify

\frac{25 !}{24 ! ( 25 - 24 ) !} (- 1)^{1} (3 i)^{24} : - 13286025 i^{24}

Simplify

\frac{25 !}{25 ! ( 25 - 25 ) !} (- 1)^{0} (3 i)^{25} : 5314413 i^{25}

= - 1 + 253 i - 900 i^{2} + 69003 i^{3} - 113850 i^{4} + 4781703 i^{5} - 4781700 i^{6} + 129789003 i^{7} - 87607575 i^{8} + 1654809753 i^{9} - 794308680 i^{10} + 10831482003 i^{11} - 3791018700 i^{12} + 37910187003 i^{13} - 9748333800 i^{14} + 71487781203 i^{15} - 13403958975 i^{16} + 70962135753 i^{17} - 9461618100 i^{18} + 34858593003 i^{19} - 3137273370 i^{20} + 7469698503 i^{21} - 407438100 i^{22} + 531441003 i^{23} - 13286025 i^{24} + 5314413 i^{25}

Simplify

- 1 + 253 i - 900 i^{2} + 69003 i^{3} - 113850 i^{4} + 4781703 i^{5} - 4781700 i^{6} + 129789003 i^{7} - 87607575 i^{8} + 1654809753 i^{9} - 794308680 i^{10} + 10831482003 i^{11} - 3791018700 i^{12} + 37910187003 i^{13} - 9748333800 i^{14} + 71487781203 i^{15} - 13403958975 i^{16} + 70962135753 i^{17} - 9461618100 i^{18} + 34858593003 i^{19} - 3137273370 i^{20} + 7469698503 i^{21} - 407438100 i^{22} + 531441003 i^{23} - 13286025 i^{24} + 5314413 i^{25} : 167772163 i - 16777216

= 167772163 i - 16777216

Rewrite in standard complex form:

- 16777216 + 167772163 i

= - 16777216 + 167772163 i

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Frequently Asked Questions (FAQ)

What is (-1+sqrt(3)i)^{25} ?
The solution to (-1+sqrt(3)i)^{25} is -16777216+16777216sqrt(3)i