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Popular >

simplify (-1+3i)^6

Solution

simplify

(- 1 + 3 i)^{6}

Solution

352 - 936 i

Solution steps

(- 1 + 3 i)^{6}

Apply binomial theorem:

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

a = - 1, b = 3 i

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

Expand summation

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

Simplify

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

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} (- 1)^{5} (3 i)^{1} : - 18 i

Simplify

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

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} (- 1)^{3} (3 i)^{3} : - 540 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} (- 1)^{2} (3 i)^{4} : 1215 i^{4}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} (- 1)^{1} (3 i)^{5} : - 1458 i^{5}

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} (- 1)^{0} (3 i)^{6} : 729 i^{6}

= 1 - 18 i + 135 i^{2} - 540 i^{3} + 1215 i^{4} - 1458 i^{5} + 729 i^{6}

Simplify

1 - 18 i + 135 i^{2} - 540 i^{3} + 1215 i^{4} - 1458 i^{5} + 729 i^{6} : - 936 i + 352

= - 936 i + 352

Rewrite in standard complex form:

352 - 936 i

= 352 - 936 i

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Frequently Asked Questions (FAQ)

What is (-1+3i)^6 ?
The solution to (-1+3i)^6 is 352-936i