Question

Simplify (Ci) `(2 - 3i) / (1 + 2i)` and (Cii)`(3i^12-i^9)/(2i+1)`?

Answer 1

`(2 - 3i) / (1 + 2i) = -4/5-7/5i`

`(3i^12-i^9)/(2i+1) = 1/5-7/5i`

Explanation:

We can remove a complex denominator by multiplying the quotient by its complex conjugate (and therefore we must also multiply the numerator), This woks because the product of a complex number with its conjugate is always real, and ia a similar way to rationalising the denominator of an irrational expression to remove a surd denominator.

Question Ci

Here the conjugate of the denominator is `1 - 2i` so we get:

`(2 - 3i) / (1 + 2i) = ((2 - 3i)) / ((1 + 2i)) * ((1 - 2i))/((1 - 2i))`
`" " = (2-3i-4i+6i^2) / (1+2i-2i-4i^2)`
`" " = (2-7i-6) / (1+4) \ \ \ \ \ (because i^2=-1)`
`" " = (-4-7i) / 5`
`" " = -4/5-7/5i`

Question Cii
Here the conjugate of the denominator is also `1 - 2i` so we get:

`(3i^12-i^9)/(2i+1) = (3-i)/(1+2i)`
`" " = ((3-i))/((1+2i)) * ((1 - 2i))/((1 - 2i))`
`" " = (3-i-6i+2i^2)/(1+2i-2i-4i^2)`
`" " = (3-7i-2)/(1+4)`
`" " = (1-7i)/5`
`" " = 1/5-7/5i`