What is (4+sqrt(6)i)^8 ?

The solution to (4+sqrt(6)i)^8 is -72944-90880sqrt(6)i

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simplify (4+sqrt(6)i)^8

simplify

(4 + 6 i)^{8}

- 72944 - 908806 i

Solution steps

(4 + 6 i)^{8}

Apply binomial theorem:

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

a = 4, b = 6 i

= i = 0 \sum 8 (i 8) \cdot 4^{(8 - i)} (6 i)^{i}

Expand summation

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

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} \cdot 4^{8} (6 i)^{0} : 65536

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} \cdot 4^{7} (6 i)^{1} : 1310726 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} \cdot 4^{6} (6 i)^{2} : 688128 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} \cdot 4^{5} (6 i)^{3} : 3440646 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} \cdot 4^{4} (6 i)^{4} : 645120 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} \cdot 4^{3} (6 i)^{5} : 1290246 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} \cdot 4^{2} (6 i)^{6} : 96768 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} \cdot 4^{1} (6 i)^{7} : 69126 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} \cdot 4^{0} (6 i)^{8} : 1296 i^{8}

= 65536 + 1310726 i + 688128 i^{2} + 3440646 i^{3} + 645120 i^{4} + 1290246 i^{5} + 96768 i^{6} + 69126 i^{7} + 1296 i^{8}

Simplify

65536 + 1310726 i + 688128 i^{2} + 3440646 i^{3} + 645120 i^{4} + 1290246 i^{5} + 96768 i^{6} + 69126 i^{7} + 1296 i^{8} : - 908806 i - 72944

= - 908806 i - 72944

Rewrite in standard complex form:

- 72944 - 908806 i

= - 72944 - 908806 i

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