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Popular >

simplify (-2-3i)^6

Solution

simplify

(- 2 - 3 i)^{6}

Solution

2035 - 828 i

Solution steps

(- 2 - 3 i)^{6}

Apply binomial theorem:

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

a = - 2, b = - 3 i

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

Expand summation

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

Simplify

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

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (- 2)^{5} (- 3 i)^{1} : 576 i

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} (- 2)^{4} (- 3 i)^{2} : 2160 i^{2}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (- 2)^{3} (- 3 i)^{3} : 4320 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (- 2)^{2} (- 3 i)^{4} : 4860 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (- 2)^{1} (- 3 i)^{5} : 2916 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (- 2)^{0} (- 3 i)^{6} : 729 i^{6}

= 64 + 576 i + 2160 i^{2} + 4320 i^{3} + 4860 i^{4} + 2916 i^{5} + 729 i^{6}

Simplify

64 + 576 i + 2160 i^{2} + 4320 i^{3} + 4860 i^{4} + 2916 i^{5} + 729 i^{6} : - 828 i + 2035

= - 828 i + 2035

Rewrite in standard complex form:

2035 - 828 i

= 2035 - 828 i

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Frequently Asked Questions (FAQ)

What is (-2-3i)^6 ?
The solution to (-2-3i)^6 is 2035-828i