Question

Convert all complex numbers to trigonometric form and then simplify the expression. Write the answer in standard form.? `((2+2i)^5 (-3+i)^3) /(sqrt 3+i)^10`

Answer 1

`(27sqrt3)/4-i27/4`

Explanation:

So this took me so many attempts and here it goes:

Find the r values using this formula:

`r=sqrt (x^2+y^2)`

Plug in:

`([2sqrt2 (cos45^o + isin45^o)]^5[3sqrt2 (cos135^o + isin135^o)]^3 )/[2(cos 30^o + isin30^o)]^10`

Power and Multiply:

`(2sqrt2)^5` see how I powered the `2sqrt2` and you do the same with the rest

`((2sqrt2)^5(cos5 times 45^o + isin5 times45^o)(3sqrt2)^3(cos 3 times 135^o +isin3 times 135^o))/(2^10(cos10 times + isin10 times 30))`

Simplify:

`(128sqrt2(cos225^o + isin225^o) 54sqrt2(cos405^o + isin405^o))/(1024(cos 300^o + isin 300^o))`

`(13824 (cos630^o + isin630^o)) /(1024(cos300^o + isin300^o))`

Divide:

`27/2` `(cos630^o-300^o + isin630^o-300^o)`

Simplify:

`27/2` `(cos330^o + isin330^o)`

Which gives your answer . . .