What is (-3+isqrt(3))^{14} ?

The solution to (-3+isqrt(3))^{14} is 17915904-17915904sqrt(3)i

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simplify (-3+isqrt(3))^{14}

simplify

(- 3 + i 3)^{14}

17915904 - 179159043 i

Solution steps

(- 3 + i 3)^{14}

Apply binomial theorem:

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

a = - 3, b = i 3

= i = 0 \sum 14 (i 14) (- 3)^{(14 - i)} (i 3)^{i}

Expand summation

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

Simplify

\frac{14 !}{0 ! ( 14 - 0 ) !} (- 3)^{14} (i 3)^{0} : 4782969

Simplify

\frac{14 !}{1 ! ( 14 - 1 ) !} (- 3)^{13} (i 3)^{1} : - 223205223 i

Simplify

\frac{14 !}{2 ! ( 14 - 2 ) !} (- 3)^{12} (i 3)^{2} : 145083393 i^{2}

Simplify

\frac{14 !}{3 ! ( 14 - 3 ) !} (- 3)^{11} (i 3)^{3} : - 1934445243 i^{3}

Simplify

\frac{14 !}{4 ! ( 14 - 4 ) !} (- 3)^{10} (i 3)^{4} : 531972441 i^{4}

Simplify

\frac{14 !}{5 ! ( 14 - 5 ) !} (- 3)^{9} (i 3)^{5} : - 3546482943 i^{5}

Simplify

\frac{14 !}{6 ! ( 14 - 6 ) !} (- 3)^{8} (i 3)^{6} : 531972441 i^{6}

Simplify

\frac{14 !}{7 ! ( 14 - 7 ) !} (- 3)^{7} (i 3)^{7} : - 2026561683 i^{7}

Simplify

\frac{14 !}{8 ! ( 14 - 8 ) !} (- 3)^{6} (i 3)^{8} : 177324147 i^{8}

Simplify

\frac{14 !}{9 ! ( 14 - 9 ) !} (- 3)^{5} (i 3)^{9} : - 394053663 i^{9}

Simplify

\frac{14 !}{10 ! ( 14 - 10 ) !} (- 3)^{4} (i 3)^{10} : 19702683 i^{10}

Simplify

\frac{14 !}{11 ! ( 14 - 11 ) !} (- 3)^{3} (i 3)^{11} : - 23882043 i^{11}

Simplify

\frac{14 !}{12 ! ( 14 - 12 ) !} (- 3)^{2} (i 3)^{12} : 597051 i^{12}

Simplify

\frac{14 !}{13 ! ( 14 - 13 ) !} (- 3)^{1} (i 3)^{13} : - 306183 i^{13}

Simplify

\frac{14 !}{14 ! ( 14 - 14 ) !} (- 3)^{0} (i 3)^{14} : 2187 i^{14}

= 4782969 - 223205223 i + 145083393 i^{2} - 1934445243 i^{3} + 531972441 i^{4} - 3546482943 i^{5} + 531972441 i^{6} - 2026561683 i^{7} + 177324147 i^{8} - 394053663 i^{9} + 19702683 i^{10} - 23882043 i^{11} + 597051 i^{12} - 306183 i^{13} + 2187 i^{14}

Simplify

4782969 - 223205223 i + 145083393 i^{2} - 1934445243 i^{3} + 531972441 i^{4} - 3546482943 i^{5} + 531972441 i^{6} - 2026561683 i^{7} + 177324147 i^{8} - 394053663 i^{9} + 19702683 i^{10} - 23882043 i^{11} + 597051 i^{12} - 306183 i^{13} + 2187 i^{14} : - 179159043 i + 17915904

= - 179159043 i + 17915904

Rewrite in standard complex form:

17915904 - 179159043 i

= 17915904 - 179159043 i

(\frac{4}{16} x + \frac{5}{16} y + \frac{3}{16} z + \frac{4}{16} q)^{4}

\frac{p + 2}{( p + 1 ) ( p - 2 ) ( p ^{2} + 4 )}

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

\frac{( k + 1 ) ^{2} ( k + 2 ) ^{2}}{4} + (k + 1)

- \frac{4}{s ^{2} ( s - 6 )}

What is (-3+isqrt(3))^{14} ?
The solution to (-3+isqrt(3))^{14} is 17915904-17915904sqrt(3)i