Question

Question #69db0

Answer 1

The result is `41-840i`.

Explanation:

Here is Pascal's Triangle:


Here is the same triangle but in the context of binomial expansion:


In the expression `(2+5i)^4`, our `a` is `2` and our `b` is `5i`. Now, we use the fourth row in the second picture and plug in our values (I color-coded each part so it is easier to follow):

`color(white)=>color(red)(a^4)+color(orange)(4a^3b)+color(green)(6a^2b^2)+color(blue)(4ab^3+b^4)`

`=>color(red)((2)^4)+color(orange)(4(2)^3(5i))+color(green)(6(2)^2(5i)^2)+color(blue)(4(2)(5i)^3)+color(purple)((5i)^4)`

`=color(red)16+color(orange)(4*8*5i)+color(green)(6*4*5^2*i^2)+color(blue)(4*2*5^3*i^3)+color(purple)(5^4*i^4)`

`=color(red)16+color(orange)(160i)+color(green)(600*i^2)+color(blue)(1000*i^2*i)+color(purple)(625*(i^2)^2)`

`=color(red)16+color(orange)(160i)+color(green)(600*-1)+color(blue)(1000*-1*i)+color(purple)(625*(-1)^2)`

`=color(red)16+color(orange)(160i)color(green)(-600)color(blue)(-1000i)+color(purple)(625*1)`

`=color(red)16color(green)(-600)+color(purple)(625)+color(orange)(160i)color(blue)(-1000i)`

`=41+color(orange)(160i)color(blue)(-1000i)`

`=41-840i`

That's the result!