What is (1-sqrt(3)i)^5 ?

The solution to (1-sqrt(3)i)^5 is 16+16sqrt(3)i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (1-sqrt(3)i)^5

simplify

(1 - 3 i)^{5}

16 + 163 i

Solution steps

(1 - 3 i)^{5}

Apply binomial theorem:

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

a = 1, b = - 3 i

= i = 0 \sum 5 (i 5) \cdot 1^{(5 - i)} (- 3 i)^{i}

Expand summation

= \frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (- 3 i)^{0} + \frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (- 3 i)^{1} + \frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (- 3 i)^{2} + \frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (- 3 i)^{3} + \frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (- 3 i)^{4} + \frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (- 3 i)^{5}

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} \cdot 1^{5} (- 3 i)^{0} : 1

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} \cdot 1^{4} (- 3 i)^{1} : - 53 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} \cdot 1^{3} (- 3 i)^{2} : 30 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} \cdot 1^{2} (- 3 i)^{3} : - 303 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} \cdot 1^{1} (- 3 i)^{4} : 45 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} \cdot 1^{0} (- 3 i)^{5} : - 93 i^{5}

= 1 - 53 i + 30 i^{2} - 303 i^{3} + 45 i^{4} - 93 i^{5}

Simplify

1 - 53 i + 30 i^{2} - 303 i^{3} + 45 i^{4} - 93 i^{5} : 163 i + 16

= 163 i + 16

Rewrite in standard complex form:

16 + 163 i

= 16 + 163 i

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

2 (x y) 3 (x y)

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

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

(1 - 3 i)^{4}