What is (1/2+(sqrt(3))/2 i)^6 ?

The solution to (1/2+(sqrt(3))/2 i)^6 is 1

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (1/2+(sqrt(3))/2 i)^6

simplify

(\frac{1}{2} + \frac{3}{2} i)^{6}

1

Solution steps

(\frac{1}{2} + \frac{3}{2} i)^{6}

Apply binomial theorem:

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

a = \frac{1}{2}, b = \frac{3}{2} i

= i = 0 \sum 6 (i 6) (\frac{1}{2})^{(6 - i)} (\frac{3}{2} i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (\frac{1}{2})^{6} (\frac{3}{2} i)^{0} : \frac{1}{64}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (\frac{1}{2})^{5} (\frac{3}{2} i)^{1} : \frac{3 3 i}{32}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (\frac{1}{2})^{4} (\frac{3}{2} i)^{2} : \frac{45 i ^{2}}{64}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (\frac{1}{2})^{3} (\frac{3}{2} i)^{3} : \frac{15 3 i ^{3}}{16}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (\frac{1}{2})^{2} (\frac{3}{2} i)^{4} : \frac{135 i ^{4}}{64}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (\frac{1}{2})^{1} (\frac{3}{2} i)^{5} : \frac{27 3 i ^{5}}{32}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (\frac{1}{2})^{0} (\frac{3}{2} i)^{6} : \frac{27 i ^{6}}{64}

= \frac{1}{64} + \frac{3 3 i}{32} + \frac{45 i ^{2}}{64} + \frac{15 3 i ^{3}}{16} + \frac{135 i ^{4}}{64} + \frac{27 3 i ^{5}}{32} + \frac{27 i ^{6}}{64}

Simplify

\frac{1}{64} + \frac{3 3 i}{32} + \frac{45 i ^{2}}{64} + \frac{15 3 i ^{3}}{16} + \frac{135 i ^{4}}{64} + \frac{27 3 i ^{5}}{32} + \frac{27 i ^{6}}{64} : 1

= 1

10 x^{2} - 20 x - 10 = 0

(- \frac{4}{7})^{4}

a^{- 3}

8 x - 4 = 2 (3 x + 6)

30 y^{2} + 10 y - 20