What is (sqrt(3)-i)^{10} ?

The solution to (sqrt(3)-i)^{10} is 512+512sqrt(3)i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (sqrt(3)-i)^{10}

simplify

(3 - i)^{10}

512 + 5123 i

Solution steps

(3 - i)^{10}

Apply binomial theorem:

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

a = 3, b = - i

= i = 0 \sum 10 (i 10) (3)^{(10 - i)} (- i)^{i}

Expand summation

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

Simplify

\frac{10 !}{0 ! ( 10 - 0 ) !} (3)^{10} (- i)^{0} : 243

Simplify

\frac{10 !}{1 ! ( 10 - 1 ) !} (3)^{9} (- i)^{1} : - 8103 i

Simplify

\frac{10 !}{2 ! ( 10 - 2 ) !} (3)^{8} (- i)^{2} : 3645 i^{2}

Simplify

\frac{10 !}{3 ! ( 10 - 3 ) !} (3)^{7} (- i)^{3} : - 32403 i^{3}

Simplify

\frac{10 !}{4 ! ( 10 - 4 ) !} (3)^{6} (- i)^{4} : 5670 i^{4}

Simplify

\frac{10 !}{5 ! ( 10 - 5 ) !} (3)^{5} (- i)^{5} : - 22683 i^{5}

Simplify

\frac{10 !}{6 ! ( 10 - 6 ) !} (3)^{4} (- i)^{6} : 1890 i^{6}

Simplify

\frac{10 !}{7 ! ( 10 - 7 ) !} (3)^{3} (- i)^{7} : - 3603 i^{7}

Simplify

\frac{10 !}{8 ! ( 10 - 8 ) !} (3)^{2} (- i)^{8} : 135 i^{8}

Simplify

\frac{10 !}{9 ! ( 10 - 9 ) !} (3)^{1} (- i)^{9} : - 103 i^{9}

Simplify

\frac{10 !}{10 ! ( 10 - 10 ) !} (3)^{0} (- i)^{10} : i^{10}

= 243 - 8103 i + 3645 i^{2} - 32403 i^{3} + 5670 i^{4} - 22683 i^{5} + 1890 i^{6} - 3603 i^{7} + 135 i^{8} - 103 i^{9} + i^{10}

Simplify

243 - 8103 i + 3645 i^{2} - 32403 i^{3} + 5670 i^{4} - 22683 i^{5} + 1890 i^{6} - 3603 i^{7} + 135 i^{8} - 103 i^{9} + i^{10} : 5123 i + 512

= 5123 i + 512

Rewrite in standard complex form:

512 + 5123 i

= 512 + 5123 i

(3 + 25) (2 - 3)

5 (lo g_{10} (x) y + lo g_{10} (z))

(\frac{4}{5} x - \frac{2}{7} y) (\frac{4}{5} x + \frac{2}{7} y)

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

\frac{s}{( s + 2 ) ( s ^{2} + 4 )}