What is (2sqrt(3)+2i)^7 ?

The solution to (2sqrt(3)+2i)^7 is -8192sqrt(3)-8192i

English

Español

Português

Français

Deutsch

Italiano

Русский

中文(简体)

한국어

日本語

Tiếng Việt

עברית

العربية

Popular >

simplify (2sqrt(3)+2i)^7

simplify

(23 + 2 i)^{7}

- 81923 - 8192 i

Solution steps

(23 + 2 i)^{7}

Apply binomial theorem:

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

a = 23, b = 2 i

= i = 0 \sum 7 (i 7) (23)^{(7 - i)} (2 i)^{i}

Expand summation

= \frac{7 !}{0 ! ( 7 - 0 ) !} (23)^{7} (2 i)^{0} + \frac{7 !}{1 ! ( 7 - 1 ) !} (23)^{6} (2 i)^{1} + \frac{7 !}{2 ! ( 7 - 2 ) !} (23)^{5} (2 i)^{2} + \frac{7 !}{3 ! ( 7 - 3 ) !} (23)^{4} (2 i)^{3} + \frac{7 !}{4 ! ( 7 - 4 ) !} (23)^{3} (2 i)^{4} + \frac{7 !}{5 ! ( 7 - 5 ) !} (23)^{2} (2 i)^{5} + \frac{7 !}{6 ! ( 7 - 6 ) !} (23)^{1} (2 i)^{6} + \frac{7 !}{7 ! ( 7 - 7 ) !} (23)^{0} (2 i)^{7}

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} (23)^{7} (2 i)^{0} : 34563

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} (23)^{6} (2 i)^{1} : 24192 i

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} (23)^{5} (2 i)^{2} : 241923 i^{2}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} (23)^{4} (2 i)^{3} : 40320 i^{3}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} (23)^{3} (2 i)^{4} : 134403 i^{4}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} (23)^{2} (2 i)^{5} : 8064 i^{5}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} (23)^{1} (2 i)^{6} : 8963 i^{6}

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} (23)^{0} (2 i)^{7} : 128 i^{7}

= 34563 + 24192 i + 241923 i^{2} + 40320 i^{3} + 134403 i^{4} + 8064 i^{5} + 8963 i^{6} + 128 i^{7}

Simplify

34563 + 24192 i + 241923 i^{2} + 40320 i^{3} + 134403 i^{4} + 8064 i^{5} + 8963 i^{6} + 128 i^{7} : - 8192 i - 81923

= - 8192 i - 81923

Rewrite in standard complex form:

- 81923 - 8192 i

= - 81923 - 8192 i

A (2 x e^{4 x} + e^{4 x} \cdot 4 x^{2})

(11 + cot (x)) (11 - cot (x)) + csc^{2} (x)

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

\frac{- 2 s ^{2} - 6}{( s ^{2} + 4 ) ( s ^{2} - 4 )}

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