en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (-1+i)^{11}

Solution

simplify

(- 1 + i)^{11}

Solution

32 + 32 i

Solution steps

(- 1 + i)^{11}

Apply binomial theorem:

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

a = - 1, b = i

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

Simplify

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

= 32 i + 32

Rewrite in standard complex form:

32 + 32 i

= 32 + 32 i

Popular Examples

expand (sin(x)+cos(x))^2 expand

(sin (x) + cos (x))^{2}

expand (17)/((1-17x)^2)expand

\frac{17}{( 1 - 17 x ) ^{2}}

expand (x^2-1)dx+xydy expand

(x^{2} - 1) d x + x y d y

expand (2s)/((s^2+1)^2)expand

\frac{2 s}{( s ^{2} + 1 ) ^{2}}

simplify (-2+i)^6 simplify

(- 2 + i)^{6}

Frequently Asked Questions (FAQ)

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