What is (sqrt(6)-i)^6 ?

The solution to (sqrt(6)-i)^6 is -235-102sqrt(6)i

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simplify (sqrt(6)-i)^6

simplify

(6 - i)^{6}

- 235 - 1026 i

Solution steps

(6 - i)^{6}

Apply binomial theorem:

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

a = 6, b = - i

= i = 0 \sum 6 (i 6) (6)^{(6 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (6)^{6} (- i)^{0} : 216

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (6)^{5} (- i)^{1} : - 2166 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (6)^{4} (- i)^{2} : 540 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (6)^{3} (- i)^{3} : - 1206 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (6)^{2} (- i)^{4} : 90 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (6)^{1} (- i)^{5} : - 66 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (6)^{0} (- i)^{6} : i^{6}

= 216 - 2166 i + 540 i^{2} - 1206 i^{3} + 90 i^{4} - 66 i^{5} + i^{6}

Simplify

216 - 2166 i + 540 i^{2} - 1206 i^{3} + 90 i^{4} - 66 i^{5} + i^{6} : - 1026 i - 235

= - 1026 i - 235

Rewrite in standard complex form:

- 235 - 1026 i

= - 235 - 1026 i

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(x)^{3}