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Popular >

simplify (-1-i)^7

Solution

simplify

(- 1 - i)^{7}

Solution

- 8 + 8 i

Solution steps

(- 1 - i)^{7}

Apply binomial theorem:

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

a = - 1, b = - i

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

Expand summation

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

Simplify

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

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

Simplify

- 1 - 7 i - 21 i^{2} - 35 i^{3} - 35 i^{4} - 21 i^{5} - 7 i^{6} - i^{7} : 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 (-1-i)^7 ?
The solution to (-1-i)^7 is -8+8i