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Popular >

simplify (-3+2i)^5

Solution

simplify

(- 3 + 2 i)^{5}

Solution

597 + 122 i

Solution steps

(- 3 + 2 i)^{5}

Apply binomial theorem:

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

a = - 3, b = 2 i

= i = 0 \sum 5 (i 5) (- 3)^{(5 - i)} (2 i)^{i}

Expand summation

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

Simplify

\frac{5 !}{0 ! ( 5 - 0 ) !} (- 3)^{5} (2 i)^{0} : - 243

Simplify

\frac{5 !}{1 ! ( 5 - 1 ) !} (- 3)^{4} (2 i)^{1} : 810 i

Simplify

\frac{5 !}{2 ! ( 5 - 2 ) !} (- 3)^{3} (2 i)^{2} : - 1080 i^{2}

Simplify

\frac{5 !}{3 ! ( 5 - 3 ) !} (- 3)^{2} (2 i)^{3} : 720 i^{3}

Simplify

\frac{5 !}{4 ! ( 5 - 4 ) !} (- 3)^{1} (2 i)^{4} : - 240 i^{4}

Simplify

\frac{5 !}{5 ! ( 5 - 5 ) !} (- 3)^{0} (2 i)^{5} : 32 i^{5}

= - 243 + 810 i - 1080 i^{2} + 720 i^{3} - 240 i^{4} + 32 i^{5}

Simplify

- 243 + 810 i - 1080 i^{2} + 720 i^{3} - 240 i^{4} + 32 i^{5} : 122 i + 597

= 122 i + 597

Rewrite in standard complex form:

597 + 122 i

= 597 + 122 i

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Frequently Asked Questions (FAQ)

What is (-3+2i)^5 ?
The solution to (-3+2i)^5 is 597+122i