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Popular >

simplify (-1+3i)^8

Solution

simplify

(- 1 + 3 i)^{8}

Solution

- 8432 + 5376 i

Solution steps

(- 1 + 3 i)^{8}

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 8 (i 8) (- 1)^{(8 - i)} (3 i)^{i}

Expand summation

= \frac{8 !}{0 ! ( 8 - 0 ) !} (- 1)^{8} (3 i)^{0} + \frac{8 !}{1 ! ( 8 - 1 ) !} (- 1)^{7} (3 i)^{1} + \frac{8 !}{2 ! ( 8 - 2 ) !} (- 1)^{6} (3 i)^{2} + \frac{8 !}{3 ! ( 8 - 3 ) !} (- 1)^{5} (3 i)^{3} + \frac{8 !}{4 ! ( 8 - 4 ) !} (- 1)^{4} (3 i)^{4} + \frac{8 !}{5 ! ( 8 - 5 ) !} (- 1)^{3} (3 i)^{5} + \frac{8 !}{6 ! ( 8 - 6 ) !} (- 1)^{2} (3 i)^{6} + \frac{8 !}{7 ! ( 8 - 7 ) !} (- 1)^{1} (3 i)^{7} + \frac{8 !}{8 ! ( 8 - 8 ) !} (- 1)^{0} (3 i)^{8}

Simplify

\frac{8 !}{0 ! ( 8 - 0 ) !} (- 1)^{8} (3 i)^{0} : 1

Simplify

\frac{8 !}{1 ! ( 8 - 1 ) !} (- 1)^{7} (3 i)^{1} : - 24 i

Simplify

\frac{8 !}{2 ! ( 8 - 2 ) !} (- 1)^{6} (3 i)^{2} : 252 i^{2}

Simplify

\frac{8 !}{3 ! ( 8 - 3 ) !} (- 1)^{5} (3 i)^{3} : - 1512 i^{3}

Simplify

\frac{8 !}{4 ! ( 8 - 4 ) !} (- 1)^{4} (3 i)^{4} : 5670 i^{4}

Simplify

\frac{8 !}{5 ! ( 8 - 5 ) !} (- 1)^{3} (3 i)^{5} : - 13608 i^{5}

Simplify

\frac{8 !}{6 ! ( 8 - 6 ) !} (- 1)^{2} (3 i)^{6} : 20412 i^{6}

Simplify

\frac{8 !}{7 ! ( 8 - 7 ) !} (- 1)^{1} (3 i)^{7} : - 17496 i^{7}

Simplify

\frac{8 !}{8 ! ( 8 - 8 ) !} (- 1)^{0} (3 i)^{8} : 6561 i^{8}

= 1 - 24 i + 252 i^{2} - 1512 i^{3} + 5670 i^{4} - 13608 i^{5} + 20412 i^{6} - 17496 i^{7} + 6561 i^{8}

Simplify

1 - 24 i + 252 i^{2} - 1512 i^{3} + 5670 i^{4} - 13608 i^{5} + 20412 i^{6} - 17496 i^{7} + 6561 i^{8} : 5376 i - 8432

= 5376 i - 8432

Rewrite in standard complex form:

- 8432 + 5376 i

= - 8432 + 5376 i

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Frequently Asked Questions (FAQ)

What is (-1+3i)^8 ?
The solution to (-1+3i)^8 is -8432+5376i