What is (sqrt(5)-i)^6 ?

The solution to (sqrt(5)-i)^6 is -176-56sqrt(5)i

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simplify (sqrt(5)-i)^6

simplify

(5 - i)^{6}

- 176 - 565 i

Solution steps

(5 - i)^{6}

Apply binomial theorem:

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

a = 5, b = - i

= i = 0 \sum 6 (i 6) (5)^{(6 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (5)^{6} (- i)^{0} : 125

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (5)^{5} (- i)^{1} : - 1505 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (5)^{4} (- i)^{2} : 375 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (5)^{3} (- i)^{3} : - 1005 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (5)^{2} (- i)^{4} : 75 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (5)^{1} (- i)^{5} : - 65 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (5)^{0} (- i)^{6} : i^{6}

= 125 - 1505 i + 375 i^{2} - 1005 i^{3} + 75 i^{4} - 65 i^{5} + i^{6}

Simplify

125 - 1505 i + 375 i^{2} - 1005 i^{3} + 75 i^{4} - 65 i^{5} + i^{6} : - 565 i - 176

= - 565 i - 176

Rewrite in standard complex form:

- 176 - 565 i

= - 176 - 565 i

\frac{( k )}{( 0.5 p + 1 ) ( 0.75 p + 1 ) ( 0.36 \cdot p + 1 )}

(3 x + 3 d x - 5) (3 x + 3 d x - 5)

x^{2} \cdot (ln (x + 2) - 1) d x

- \frac{3}{4} (x + 2) (x + 1) (x - 3)

\frac{20 ( 5 + 2 5 )}{3}