What is (sqrt(3)+i)^7 ?

The solution to (sqrt(3)+i)^7 is -64sqrt(3)-64i

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simplify (sqrt(3)+i)^7

simplify

(3 + i)^{7}

- 643 - 64 i

Solution steps

(3 + i)^{7}

Apply binomial theorem:

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

a = 3, b = i

= i = 0 \sum 7 (i 7) (3)^{(7 - i)} i^{i}

Expand summation

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

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} (3)^{7} i^{0} : 273

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} (3)^{6} i^{1} : 189 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} (3)^{5} i^{2} : 1893 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} (3)^{4} i^{3} : 315 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} (3)^{3} i^{4} : 1053 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} (3)^{2} i^{5} : 63 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} (3)^{1} i^{6} : 73 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} (3)^{0} i^{7} : i^{7}

= 273 + 189 i + 1893 i^{2} + 315 i^{3} + 1053 i^{4} + 63 i^{5} + 73 i^{6} + i^{7}

Simplify

273 + 189 i + 1893 i^{2} + 315 i^{3} + 1053 i^{4} + 63 i^{5} + 73 i^{6} + i^{7} : - 64 i - 643

= - 64 i - 643

Rewrite in standard complex form:

- 643 - 64 i

= - 643 - 64 i

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