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Popular >

simplify (1+sqrt(3)i)^9

Solution

simplify

(1 + 3 i)^{9}

Solution

- 512

Solution steps

(1 + 3 i)^{9}

Apply binomial theorem:

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

a = 1, b = 3 i

= i = 0 \sum 9 (i 9) \cdot 1^{(9 - i)} (3 i)^{i}

Expand summation

= \frac{9 !}{0 ! ( 9 - 0 ) !} \cdot 1^{9} (3 i)^{0} + \frac{9 !}{1 ! ( 9 - 1 ) !} \cdot 1^{8} (3 i)^{1} + \frac{9 !}{2 ! ( 9 - 2 ) !} \cdot 1^{7} (3 i)^{2} + \frac{9 !}{3 ! ( 9 - 3 ) !} \cdot 1^{6} (3 i)^{3} + \frac{9 !}{4 ! ( 9 - 4 ) !} \cdot 1^{5} (3 i)^{4} + \frac{9 !}{5 ! ( 9 - 5 ) !} \cdot 1^{4} (3 i)^{5} + \frac{9 !}{6 ! ( 9 - 6 ) !} \cdot 1^{3} (3 i)^{6} + \frac{9 !}{7 ! ( 9 - 7 ) !} \cdot 1^{2} (3 i)^{7} + \frac{9 !}{8 ! ( 9 - 8 ) !} \cdot 1^{1} (3 i)^{8} + \frac{9 !}{9 ! ( 9 - 9 ) !} \cdot 1^{0} (3 i)^{9}

Simplify

\frac{9 !}{0 ! ( 9 - 0 ) !} \cdot 1^{9} (3 i)^{0} : 1

Simplify

\frac{9 !}{1 ! ( 9 - 1 ) !} \cdot 1^{8} (3 i)^{1} : 93 i

Simplify

\frac{9 !}{2 ! ( 9 - 2 ) !} \cdot 1^{7} (3 i)^{2} : 108 i^{2}

Simplify

\frac{9 !}{3 ! ( 9 - 3 ) !} \cdot 1^{6} (3 i)^{3} : 2523 i^{3}

Simplify

\frac{9 !}{4 ! ( 9 - 4 ) !} \cdot 1^{5} (3 i)^{4} : 1134 i^{4}

Simplify

\frac{9 !}{5 ! ( 9 - 5 ) !} \cdot 1^{4} (3 i)^{5} : 11343 i^{5}

Simplify

\frac{9 !}{6 ! ( 9 - 6 ) !} \cdot 1^{3} (3 i)^{6} : 2268 i^{6}

Simplify

\frac{9 !}{7 ! ( 9 - 7 ) !} \cdot 1^{2} (3 i)^{7} : 9723 i^{7}

Simplify

\frac{9 !}{8 ! ( 9 - 8 ) !} \cdot 1^{1} (3 i)^{8} : 729 i^{8}

Simplify

\frac{9 !}{9 ! ( 9 - 9 ) !} \cdot 1^{0} (3 i)^{9} : 813 i^{9}

= 1 + 93 i + 108 i^{2} + 2523 i^{3} + 1134 i^{4} + 11343 i^{5} + 2268 i^{6} + 9723 i^{7} + 729 i^{8} + 813 i^{9}

Simplify

1 + 93 i + 108 i^{2} + 2523 i^{3} + 1134 i^{4} + 11343 i^{5} + 2268 i^{6} + 9723 i^{7} + 729 i^{8} + 813 i^{9} : - 512

= - 512

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Frequently Asked Questions (FAQ)

What is (1+sqrt(3)i)^9 ?
The solution to (1+sqrt(3)i)^9 is -512