What is (sqrt(5)-i)^4 ?

The solution to (sqrt(5)-i)^4 is -4-16sqrt(5)i

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simplify (sqrt(5)-i)^4

simplify

(5 - i)^{4}

- 4 - 165 i

Solution steps

(5 - i)^{4}

Apply binomial theorem:

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

a = 5, b = - i

= i = 0 \sum 4 (i 4) (5)^{(4 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (5)^{4} (- i)^{0} : 25

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (5)^{3} (- i)^{1} : - 205 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (5)^{2} (- i)^{2} : 30 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (5)^{1} (- i)^{3} : - 45 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (5)^{0} (- i)^{4} : i^{4}

= 25 - 205 i + 30 i^{2} - 45 i^{3} + i^{4}

Simplify

25 - 205 i + 30 i^{2} - 45 i^{3} + i^{4} : - 165 i - 4

= - 165 i - 4

Rewrite in standard complex form:

- 4 - 165 i

= - 4 - 165 i

(x + \frac{5}{4})^{2}

3 l e f t (m + 1) - 2

(4 a + b) (4 a - b)

(x + 2 - 1)^{2}