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Popular >

simplify (-2+4i)^6

Solution

simplify

(- 2 + 4 i)^{6}

Solution

7488 - 2816 i

Solution steps

(- 2 + 4 i)^{6}

Apply binomial theorem:

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

a = - 2, b = 4 i

= i = 0 \sum 6 (i 6) (- 2)^{(6 - i)} (4 i)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} (- 2)^{6} (4 i)^{0} : 64

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (- 2)^{5} (4 i)^{1} : - 768 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (- 2)^{4} (4 i)^{2} : 3840 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (- 2)^{3} (4 i)^{3} : - 10240 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (- 2)^{2} (4 i)^{4} : 15360 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (- 2)^{1} (4 i)^{5} : - 12288 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (- 2)^{0} (4 i)^{6} : 4096 i^{6}

= 64 - 768 i + 3840 i^{2} - 10240 i^{3} + 15360 i^{4} - 12288 i^{5} + 4096 i^{6}

Simplify

64 - 768 i + 3840 i^{2} - 10240 i^{3} + 15360 i^{4} - 12288 i^{5} + 4096 i^{6} : - 2816 i + 7488

= - 2816 i + 7488

Rewrite in standard complex form:

7488 - 2816 i

= 7488 - 2816 i

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Frequently Asked Questions (FAQ)

What is (-2+4i)^6 ?
The solution to (-2+4i)^6 is 7488-2816i