en

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (1-i)^{10}

Solution

simplify

(1 - i)^{10}

Solution

- 32 i

Solution steps

(1 - i)^{10}

Apply binomial theorem:

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

a = 1, b = - i

= i = 0 \sum 10 (i 10) \cdot 1^{(10 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} \cdot 1^{10} (- i)^{0} : 1

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} \cdot 1^{9} (- i)^{1} : - 10 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} \cdot 1^{8} (- i)^{2} : 45 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} \cdot 1^{7} (- i)^{3} : - 120 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} \cdot 1^{6} (- i)^{4} : 210 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} \cdot 1^{5} (- i)^{5} : - 252 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} \cdot 1^{4} (- i)^{6} : 210 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} \cdot 1^{3} (- i)^{7} : - 120 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} \cdot 1^{2} (- i)^{8} : 45 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} \cdot 1^{1} (- i)^{9} : - 10 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} \cdot 1^{0} (- i)^{10} : i^{10}

= 1 - 10 i + 45 i^{2} - 120 i^{3} + 210 i^{4} - 252 i^{5} + 210 i^{6} - 120 i^{7} + 45 i^{8} - 10 i^{9} + i^{10}

Simplify

1 - 10 i + 45 i^{2} - 120 i^{3} + 210 i^{4} - 252 i^{5} + 210 i^{6} - 120 i^{7} + 45 i^{8} - 10 i^{9} + i^{10} : - 32 i

= - 32 i

Popular Examples

\frac{x}{2} = \frac{6}{4}

simplify \sqrt[3]{a^6}simplify

0 = x^{2} + 2 x - 8

factor 2r^2-6r+12-4r factor

2 r^{2} - 6 r + 12 - 4 r

solvefor y,x^{-2}=(y^2)/2+1/2 ln(x)+c solve for

y, x^{- 2} = \frac{y ^{2}}{2} + \frac{1}{2} ln (x) + c

Frequently Asked Questions (FAQ)

What is (1-i)^{10} ?
The solution to (1-i)^{10} is -32i