Express in the form of a+ib

Question

(3+i√5)(3-i√5)÷(√3+√2i)-(√3-i√2)

Answer 1

If your expression mean:

( 3 + i √5 ) ( 3 - i √5 ) / [ ( √3 + √2 i ) - ( √3 - i √2 ) ]

then:

[ ( 3 + i √5 ) ∙ ( 3 - i √5 ) ] / [ ( √3 + √2 i ) - ( √3 - i √2 ) ] =

[ 3 ∙ 3 + 3 ∙ i √5 - i √5 ∙ 3 - i √5 ∙ i √5 ] / [ √3 + √2 i - √3 - ( - √2 i ) ] =

[ 9 + 3 i √5 - 3 i √5 - ( i √5 )² ] / [ √3 + √2 i - √3 + √2 i ] =

( 9 - i² ∙ √5² ) / [ √3 - √3 + √2 i + √2 i ] =

( 9 - ( - 1 ) ∙ 5 ) / ( √2 i + √2 i ) =

[ 9 - ( - 5 ) ] / 2 √2 i =

( 9 + 5 ) / 2 √2 i =

14 / 2 √2 i =

2 ∙ 7 / 2 √2 i =

7 / √2 i =

7 ∙ i / √2 i ∙ i =

7 i / √2 i² =

7 i / √2 ∙ ( - 1 ) =

7 i / - √2 =

- 7 i / √2 =

0 - 7 i / √2 =

0 - i 7 / √2

( 3 + i √5 ) ( 3 - i √5 ) / [ ( √3 + √2 i ) - ( √3 - i √2 ) ] = 0 - i 7 / √2

Answer 2

Can you see the confusion created by insufficient use of brackets ?

I will take it as face value the way you typed it ....

(3+i√5)(3-i√5)÷(√3+√2i)-(√3-i√2) ------> why is i in front of -i√2, but behind in √2i ??
= (9 - 5i^2) ÷ (√3+i√2) - (√3-i√2)
= 14/(√3+i√2) - (√3-i√2)
= 14/(√3+i√2)*( (√3-i√2)/(√3-i√2) ) - (√3-i√2)
= 14(√3 - i√2)/5 - (√3 - i√2)
= 14(√3 - i√2)/5 - 5(√3 - i√2)/5
= (9/5)(√3 - i√2)