Question

Simplify the expression fraction with numerator of the square root of negative four and denominator of the quantity three plus i minus the quantity two plus three times i. HINT: Simplify the denominator first. Then use the conjugate of the denominator to rationalize the fraction

Answer 1

`(-4+2i)/(5)`

Explanation:

We are given the expression,
`(√(-4))/((3+i)-(2+3i))`

On simplifying, we have,

`(2i)/(3+i-2-3i)`

i.e.
`(2i)/(1-2i)`

Now, we will rationalize the expression,

i.e.
`(2i)/(1-2i)* (1+2i)/(1+2i)`

i.e.
`((2i)* (1+2i))/((1-2i)* (1+2i))`

i.e.
`(2i+4i^(2))/(1-4i^(2))`

Since,
`i^(2)=-1`, we get,

i.e.
`(2i-4)/(1+4)`

i.e.
`(-4+2i)/(5)`

So, the simplified expression is
`(-4+2i)/(5)`.