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Popular >

simplify (1-i)^{13}

Solution

simplify

(1 - i)^{13}

Solution

- 64 + 64 i

Solution steps

(1 - i)^{13}

Apply binomial theorem:

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

a = 1, b = - i

= i = 0 \sum 13 (i 13) \cdot 1^{(13 - i)} (- i)^{i}

Expand summation

= \frac{13 !}{0 ! ( 13 - 0 ) !} \cdot 1^{13} (- i)^{0} + \frac{13 !}{1 ! ( 13 - 1 ) !} \cdot 1^{12} (- i)^{1} + \frac{13 !}{2 ! ( 13 - 2 ) !} \cdot 1^{11} (- i)^{2} + \frac{13 !}{3 ! ( 13 - 3 ) !} \cdot 1^{10} (- i)^{3} + \frac{13 !}{4 ! ( 13 - 4 ) !} \cdot 1^{9} (- i)^{4} + \frac{13 !}{5 ! ( 13 - 5 ) !} \cdot 1^{8} (- i)^{5} + \frac{13 !}{6 ! ( 13 - 6 ) !} \cdot 1^{7} (- i)^{6} + \frac{13 !}{7 ! ( 13 - 7 ) !} \cdot 1^{6} (- i)^{7} + \frac{13 !}{8 ! ( 13 - 8 ) !} \cdot 1^{5} (- i)^{8} + \frac{13 !}{9 ! ( 13 - 9 ) !} \cdot 1^{4} (- i)^{9} + \frac{13 !}{10 ! ( 13 - 10 ) !} \cdot 1^{3} (- i)^{10} + \frac{13 !}{11 ! ( 13 - 11 ) !} \cdot 1^{2} (- i)^{11} + \frac{13 !}{12 ! ( 13 - 12 ) !} \cdot 1^{1} (- i)^{12} + \frac{13 !}{13 ! ( 13 - 13 ) !} \cdot 1^{0} (- i)^{13}

Simplify

\frac{13 !}{0 ! ( 13 - 0 ) !} \cdot 1^{13} (- i)^{0} : 1

Simplify

\frac{13 !}{1 ! ( 13 - 1 ) !} \cdot 1^{12} (- i)^{1} : - 13 i

Simplify

\frac{13 !}{2 ! ( 13 - 2 ) !} \cdot 1^{11} (- i)^{2} : 78 i^{2}

Simplify

\frac{13 !}{3 ! ( 13 - 3 ) !} \cdot 1^{10} (- i)^{3} : - 286 i^{3}

Simplify

\frac{13 !}{4 ! ( 13 - 4 ) !} \cdot 1^{9} (- i)^{4} : 715 i^{4}

Simplify

\frac{13 !}{5 ! ( 13 - 5 ) !} \cdot 1^{8} (- i)^{5} : - 1287 i^{5}

Simplify

\frac{13 !}{6 ! ( 13 - 6 ) !} \cdot 1^{7} (- i)^{6} : 1716 i^{6}

Simplify

\frac{13 !}{7 ! ( 13 - 7 ) !} \cdot 1^{6} (- i)^{7} : - 1716 i^{7}

Simplify

\frac{13 !}{8 ! ( 13 - 8 ) !} \cdot 1^{5} (- i)^{8} : 1287 i^{8}

Simplify

\frac{13 !}{9 ! ( 13 - 9 ) !} \cdot 1^{4} (- i)^{9} : - 715 i^{9}

Simplify

\frac{13 !}{10 ! ( 13 - 10 ) !} \cdot 1^{3} (- i)^{10} : 286 i^{10}

Simplify

\frac{13 !}{11 ! ( 13 - 11 ) !} \cdot 1^{2} (- i)^{11} : - 78 i^{11}

Simplify

\frac{13 !}{12 ! ( 13 - 12 ) !} \cdot 1^{1} (- i)^{12} : 13 i^{12}

Simplify

\frac{13 !}{13 ! ( 13 - 13 ) !} \cdot 1^{0} (- i)^{13} : - i^{13}

= 1 - 13 i + 78 i^{2} - 286 i^{3} + 715 i^{4} - 1287 i^{5} + 1716 i^{6} - 1716 i^{7} + 1287 i^{8} - 715 i^{9} + 286 i^{10} - 78 i^{11} + 13 i^{12} - i^{13}

Simplify

1 - 13 i + 78 i^{2} - 286 i^{3} + 715 i^{4} - 1287 i^{5} + 1716 i^{6} - 1716 i^{7} + 1287 i^{8} - 715 i^{9} + 286 i^{10} - 78 i^{11} + 13 i^{12} - i^{13} : 64 i - 64

= 64 i - 64

Rewrite in standard complex form:

- 64 + 64 i

= - 64 + 64 i

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Frequently Asked Questions (FAQ)

What is (1-i)^{13} ?
The solution to (1-i)^{13} is -64+64i