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Popular >

simplify (-1+i)^5

Solution

simplify

(- 1 + i)^{5}

Solution

4 - 4 i

Solution steps

(- 1 + i)^{5}

Apply binomial theorem:

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

a = - 1, b = i

= i = 0 \sum 5 (i 5) (- 1)^{(5 - i)} i^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (- 1)^{5} i^{0} : - 1

Simplify

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

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (- 1)^{3} i^{2} : - 10 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (- 1)^{2} i^{3} : 10 i^{3}

Simplify

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

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (- 1)^{0} i^{5} : i^{5}

= - 1 + 5 i - 10 i^{2} + 10 i^{3} - 5 i^{4} + i^{5}

Simplify

- 1 + 5 i - 10 i^{2} + 10 i^{3} - 5 i^{4} + i^{5} : - 4 i + 4

= - 4 i + 4

Rewrite in standard complex form:

4 - 4 i

= 4 - 4 i

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Frequently Asked Questions (FAQ)

What is (-1+i)^5 ?
The solution to (-1+i)^5 is 4-4i