What is (sqrt(5)-i)^8 ?

The solution to (sqrt(5)-i)^8 is -1264+128sqrt(5)i

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simplify (sqrt(5)-i)^8

simplify

(5 - i)^{8}

- 1264 + 1285 i

Solution steps

(5 - i)^{8}

Apply binomial theorem:

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

a = 5, b = - i

= i = 0 \sum 8 (i 8) (5)^{(8 - i)} (- i)^{i}

Expand summation

= \frac{8 !}{0 ! ( 8 - 0 ) !} (5)^{8} (- i)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} (5)^{7} (- i)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} (5)^{6} (- i)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} (5)^{5} (- i)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} (5)^{4} (- i)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} (5)^{3} (- i)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} (5)^{2} (- i)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} (5)^{1} (- i)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} (5)^{0} (- i)^{8}

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} (5)^{8} (- i)^{0} : 625

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} (5)^{7} (- i)^{1} : - 10005 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} (5)^{6} (- i)^{2} : 3500 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} (5)^{5} (- i)^{3} : - 14005 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} (5)^{4} (- i)^{4} : 1750 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} (5)^{3} (- i)^{5} : - 2805 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} (5)^{2} (- i)^{6} : 140 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} (5)^{1} (- i)^{7} : - 85 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} (5)^{0} (- i)^{8} : i^{8}

= 625 - 10005 i + 3500 i^{2} - 14005 i^{3} + 1750 i^{4} - 2805 i^{5} + 140 i^{6} - 85 i^{7} + i^{8}

Simplify

625 - 10005 i + 3500 i^{2} - 14005 i^{3} + 1750 i^{4} - 2805 i^{5} + 140 i^{6} - 85 i^{7} + i^{8} : 1285 i - 1264

= 1285 i - 1264

Rewrite in standard complex form:

- 1264 + 1285 i

= - 1264 + 1285 i

(\frac{3}{2 x} + \frac{1}{y})^{6}

\frac{1 - e ^{- x}}{x ^{2} ( x + 2 )}

(1 - 1 - x^{2})^{2}

4 (x - \frac{1}{2})

\frac{x ^{2} + 4 x - 2}{( x + 3 ) ( x - 2 )}