What is (sqrt(3)+i)^8 ?

The solution to (sqrt(3)+i)^8 is -128-128sqrt(3)i

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simplify (sqrt(3)+i)^8

simplify

(3 + i)^{8}

- 128 - 1283 i

Solution steps

(3 + i)^{8}

Apply binomial theorem:

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

a = 3, b = i

= i = 0 \sum 8 (i 8) (3)^{(8 - i)} i^{i}

Expand summation

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

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} (3)^{8} i^{0} : 81

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} (3)^{7} i^{1} : 2163 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} (3)^{6} i^{2} : 756 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} (3)^{5} i^{3} : 5043 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} (3)^{4} i^{4} : 630 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} (3)^{3} i^{5} : 1683 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} (3)^{2} i^{6} : 84 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} (3)^{1} i^{7} : 83 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} (3)^{0} i^{8} : i^{8}

= 81 + 2163 i + 756 i^{2} + 5043 i^{3} + 630 i^{4} + 1683 i^{5} + 84 i^{6} + 83 i^{7} + i^{8}

Simplify

81 + 2163 i + 756 i^{2} + 5043 i^{3} + 630 i^{4} + 1683 i^{5} + 84 i^{6} + 83 i^{7} + i^{8} : - 1283 i - 128

= - 1283 i - 128

Rewrite in standard complex form:

- 128 - 1283 i

= - 128 - 1283 i

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