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Popular >

simplify (i-1)^{11}

Solution

simplify

(i - 1)^{11}

Solution

32 + 32 i

Solution steps

(i - 1)^{11}

Apply binomial theorem:

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

a = i, b = - 1

= i = 0 \sum 11 (i 11) i^{(11 - i)} (- 1)^{i}

Expand summation

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

Simplify

\frac{11 !}{0 ! ( 11 - 0 ) !} i^{11} (- 1)^{0} : i^{11}

Simplify

\frac{11 !}{1 ! ( 11 - 1 ) !} i^{10} (- 1)^{1} : - 11 i^{10}

Simplify

\frac{11 !}{2 ! ( 11 - 2 ) !} i^{9} (- 1)^{2} : 55 i^{9}

Simplify

\frac{11 !}{3 ! ( 11 - 3 ) !} i^{8} (- 1)^{3} : - 165 i^{8}

Simplify

\frac{11 !}{4 ! ( 11 - 4 ) !} i^{7} (- 1)^{4} : 330 i^{7}

Simplify

\frac{11 !}{5 ! ( 11 - 5 ) !} i^{6} (- 1)^{5} : - 462 i^{6}

Simplify

\frac{11 !}{6 ! ( 11 - 6 ) !} i^{5} (- 1)^{6} : 462 i^{5}

Simplify

\frac{11 !}{7 ! ( 11 - 7 ) !} i^{4} (- 1)^{7} : - 330 i^{4}

Simplify

\frac{11 !}{8 ! ( 11 - 8 ) !} i^{3} (- 1)^{8} : 165 i^{3}

Simplify

\frac{11 !}{9 ! ( 11 - 9 ) !} i^{2} (- 1)^{9} : - 55 i^{2}

Simplify

\frac{11 !}{10 ! ( 11 - 10 ) !} i^{1} (- 1)^{10} : 11 i

Simplify

\frac{11 !}{11 ! ( 11 - 11 ) !} i^{0} (- 1)^{11} : - 1

= i^{11} - 11 i^{10} + 55 i^{9} - 165 i^{8} + 330 i^{7} - 462 i^{6} + 462 i^{5} - 330 i^{4} + 165 i^{3} - 55 i^{2} + 11 i - 1

Simplify

i^{11} - 11 i^{10} + 55 i^{9} - 165 i^{8} + 330 i^{7} - 462 i^{6} + 462 i^{5} - 330 i^{4} + 165 i^{3} - 55 i^{2} + 11 i - 1 : 32 i + 32

= 32 i + 32

Rewrite in standard complex form:

32 + 32 i

= 32 + 32 i

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Frequently Asked Questions (FAQ)

What is (i-1)^{11} ?
The solution to (i-1)^{11} is 32+32i