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) \cdot 1^{(11 - i)} (- i)^{i}

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

\frac{11 !}{11 ! ( 11 - 11 ) !} \cdot 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 x^{2/3}(1-2x)expand

x^{\frac{2}{3}} (1 - 2 x)

expand (x-y)dx+(x+y)dy expand

(x - y) d x + (x + y) d y

expand 1/((x^2+1)(x^2+4))expand

\frac{1}{( x ^{2} + 1 ) ( x ^{2} + 4 )}

simplify (x-sqrt(1+x^2))^2 simplify

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

expand (cos(4t)+cos(3t))^2 expand

(cos (4 t) + cos (3 t))^{2}

Frequently Asked Questions (FAQ)

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