Question

4i8-3i9+3÷3i11 - 4i10-2​

Answer 1

Answer:

23/13 + 15i/13

Step-by-step explanation:

we are given that

( 4i^8 - 3i^9 + 3 ) / ( 3i^11 - 4i^10 - 2 )

Since

i^8 = (i²)^4 = (-1)^4 = 1

i^9 = i^(8+1) = ( i^8 ) i = 1 i  = i

i^11 = i^( 8 + 3 ) = (i^8)(i³) = 1i³ = (i²)i = -i

i^10 = i^(9+1) = ( i^9 )(i) = (i)i = i² = -1

So

  ( 4i^8 - 3i^9 + 3 ) / ( 3i^11 - 4i^10 - 2 )

= ( 4 - 3i + 3 ) / ( 3(-i) - 4(-1) - 2 )

= ( -3i + 7 ) / ( -3i + 4 -2 )

= ( 7 - 3i ) / ( 2 - 3i )

By rationalization we get

= ( 7 - 3i )( 2 + 3i ) / ( 2 - 3i )( 2 + 3i )

= ( 14 + 21i - 6i - 9i² ) / ( 4 + 9 )

= ( 23 + 15i ) / 13

= 23/13 + 15i/13  

Thus

( 4i^8 - 3i^9 + 3 ) / ( 3i^11 - 4i^10 - 2 ) = 23/13 + 15i/13