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Popular >

simplify (sqrt(3)-i)^9

Solution

simplify

(3 - i)^{9}

Solution

512 i

Solution steps

(3 - i)^{9}

Apply binomial theorem:

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

a = 3, b = - i

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

Simplify

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

= 512 i

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Frequently Asked Questions (FAQ)

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