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Popular >

simplify (i-4)^6

Solution

simplify

(i - 4)^{6}

Solution

495 - 4888 i

Solution steps

(i - 4)^{6}

Apply binomial theorem:

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

a = i, b = - 4

= i = 0 \sum 6 (i 6) i^{(6 - i)} (- 4)^{i}

Expand summation

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

Simplify

\frac{6 !}{0 ! ( 6 - 0 ) !} i^{6} (- 4)^{0} : i^{6}

Simplify

\frac{6 !}{1 ! ( 6 - 1 ) !} i^{5} (- 4)^{1} : - 24 i^{5}

Simplify

\frac{6 !}{2 ! ( 6 - 2 ) !} i^{4} (- 4)^{2} : 240 i^{4}

Simplify

\frac{6 !}{3 ! ( 6 - 3 ) !} i^{3} (- 4)^{3} : - 1280 i^{3}

Simplify

\frac{6 !}{4 ! ( 6 - 4 ) !} i^{2} (- 4)^{4} : 3840 i^{2}

Simplify

\frac{6 !}{5 ! ( 6 - 5 ) !} i^{1} (- 4)^{5} : - 6144 i

Simplify

\frac{6 !}{6 ! ( 6 - 6 ) !} i^{0} (- 4)^{6} : 4096

= i^{6} - 24 i^{5} + 240 i^{4} - 1280 i^{3} + 3840 i^{2} - 6144 i + 4096

Simplify

i^{6} - 24 i^{5} + 240 i^{4} - 1280 i^{3} + 3840 i^{2} - 6144 i + 4096 : - 4888 i + 495

= - 4888 i + 495

Rewrite in standard complex form:

495 - 4888 i

= 495 - 4888 i

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Frequently Asked Questions (FAQ)

What is (i-4)^6 ?
The solution to (i-4)^6 is 495-4888i