Question

9i/1-3i, I know you have to mult. By conjugate I just don't know how to mult. The bottom.

Answer 1

You get the conjugate of a binomial (two terms) by multiplying the second term by -1.
For example, the conjugate of x + y is x - y. In other words I change the sign of the second term.

The conjugate is useful because when I multiply a binomial and its conjugate, for example (x + y)(x - y), I get
`x^2 - xy + xy - y^2 = x^2 - y^2`. The middle two terms xy and -xy cancel.
You may recognize the final result
`x^2 - y^2` as a difference of squares, in which the factored form is (x + y)(x - y).

The conjugate is especially helpful for simplifying fractions with imaginary numbers i (i is the square root of -1) because when the second term of a binomial has an i, you can multiply the binomial by the conjugate, in which the i will be squared, and
`i^2 = √(-1) ^2 = -1`, and of course -1 is more simplified and easier to deal with than i.


`(9i)/(1-3i) * (1+3i)/(1+3i) = (9i + 27i^2)/(1-9i^2) = (9i-27)/(1+9) = (9i-27)/(10)`