What is (sqrt(13)-i)^4 ?

The solution to (sqrt(13)-i)^4 is 92-48sqrt(13)i

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simplify (sqrt(13)-i)^4

simplify

(13 - i)^{4}

92 - 4813 i

Solution steps

(13 - i)^{4}

Apply binomial theorem:

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

a = 13, b = - i

= i = 0 \sum 4 (i 4) (13)^{(4 - i)} (- i)^{i}

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} (13)^{4} (- i)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (13)^{3} (- i)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (13)^{2} (- i)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (13)^{1} (- i)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (13)^{0} (- i)^{4}

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (13)^{4} (- i)^{0} : 169

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (13)^{3} (- i)^{1} : - 5213 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (13)^{2} (- i)^{2} : 78 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (13)^{1} (- i)^{3} : - 413 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (13)^{0} (- i)^{4} : i^{4}

= 169 - 5213 i + 78 i^{2} - 413 i^{3} + i^{4}

Simplify

169 - 5213 i + 78 i^{2} - 413 i^{3} + i^{4} : - 4813 i + 92

= - 4813 i + 92

Rewrite in standard complex form:

92 - 4813 i

= 92 - 4813 i

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x - 2^{2}