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Popular >

simplify (i-1)^5

Solution

simplify

(i - 1)^{5}

Solution

4 - 4 i

Solution steps

(i - 1)^{5}

Apply binomial theorem:

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

a = i, b = - 1

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

Simplify

i^{5} - 5 i^{4} + 10 i^{3} - 10 i^{2} + 5 i - 1 : - 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 (i-1)^5 ?
The solution to (i-1)^5 is 4-4i