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Popular >

simplify (i-1)^7

Solution

simplify

(i - 1)^{7}

Solution

- 8 - 8 i

Solution steps

(i - 1)^{7}

Apply binomial theorem:

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

a = i, b = - 1

= i = 0 \sum 7 (i 7) i^{(7 - i)} (- 1)^{i}

Expand summation

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

Simplify

\frac{7 !}{0 ! ( 7 - 0 ) !} i^{7} (- 1)^{0} : i^{7}

Simplify

\frac{7 !}{1 ! ( 7 - 1 ) !} i^{6} (- 1)^{1} : - 7 i^{6}

Simplify

\frac{7 !}{2 ! ( 7 - 2 ) !} i^{5} (- 1)^{2} : 21 i^{5}

Simplify

\frac{7 !}{3 ! ( 7 - 3 ) !} i^{4} (- 1)^{3} : - 35 i^{4}

Simplify

\frac{7 !}{4 ! ( 7 - 4 ) !} i^{3} (- 1)^{4} : 35 i^{3}

Simplify

\frac{7 !}{5 ! ( 7 - 5 ) !} i^{2} (- 1)^{5} : - 21 i^{2}

Simplify

\frac{7 !}{6 ! ( 7 - 6 ) !} i^{1} (- 1)^{6} : 7 i

Simplify

\frac{7 !}{7 ! ( 7 - 7 ) !} i^{0} (- 1)^{7} : - 1

= i^{7} - 7 i^{6} + 21 i^{5} - 35 i^{4} + 35 i^{3} - 21 i^{2} + 7 i - 1

Simplify

i^{7} - 7 i^{6} + 21 i^{5} - 35 i^{4} + 35 i^{3} - 21 i^{2} + 7 i - 1 : - 8 i - 8

= - 8 i - 8

Rewrite in standard complex form:

- 8 - 8 i

= - 8 - 8 i

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Frequently Asked Questions (FAQ)

What is (i-1)^7 ?
The solution to (i-1)^7 is -8-8i