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Popular >

simplify (-5+12i)^4

Solution

simplify

(- 5 + 12 i)^{4}

Solution

- 239 + 28560 i

Solution steps

(- 5 + 12 i)^{4}

Apply binomial theorem:

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

a = - 5, b = 12 i

= i = 0 \sum 4 (i 4) (- 5)^{(4 - i)} (12 i)^{i}

Expand summation

= \frac{4 !}{0 ! ( 4 - 0 ) !} (- 5)^{4} (12 i)^{0} + \frac{4 !}{1 ! ( 4 - 1 ) !} (- 5)^{3} (12 i)^{1} + \frac{4 !}{2 ! ( 4 - 2 ) !} (- 5)^{2} (12 i)^{2} + \frac{4 !}{3 ! ( 4 - 3 ) !} (- 5)^{1} (12 i)^{3} + \frac{4 !}{4 ! ( 4 - 4 ) !} (- 5)^{0} (12 i)^{4}

Simplify

\frac{4 !}{0 ! ( 4 - 0 ) !} (- 5)^{4} (12 i)^{0} : 625

Simplify

\frac{4 !}{1 ! ( 4 - 1 ) !} (- 5)^{3} (12 i)^{1} : - 6000 i

Simplify

\frac{4 !}{2 ! ( 4 - 2 ) !} (- 5)^{2} (12 i)^{2} : 21600 i^{2}

Simplify

\frac{4 !}{3 ! ( 4 - 3 ) !} (- 5)^{1} (12 i)^{3} : - 34560 i^{3}

Simplify

\frac{4 !}{4 ! ( 4 - 4 ) !} (- 5)^{0} (12 i)^{4} : 20736 i^{4}

= 625 - 6000 i + 21600 i^{2} - 34560 i^{3} + 20736 i^{4}

Simplify

625 - 6000 i + 21600 i^{2} - 34560 i^{3} + 20736 i^{4} : 28560 i - 239

= 28560 i - 239

Rewrite in standard complex form:

- 239 + 28560 i

= - 239 + 28560 i

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Frequently Asked Questions (FAQ)

What is (-5+12i)^4 ?
The solution to (-5+12i)^4 is -239+28560i