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

Answer:

$\frac{-4+2i}{5}$

Step-by-step explanation:

We are given the expression, $\frac{\sqrt{-4}}{(3+i)-(2+3i)}$

On simplifying, we have,

$\frac{2i}{3+i-2-3i}$

i.e. $\frac{2i}{1-2i}$

Now, we will rationalize the expression,

i.e. $\frac{2i}{1-2i}\times \frac{1+2i}{1+2i}$

i.e. $\frac{(2i)\times (1+2i)}{(1-2i)\times (1+2i)}$

i.e. $\frac{2i+4i^{2}}{1-4i^{2}}$

Since, $i^{2}=-1$, we get,

i.e. $\frac{2i-4}{1+4}$

i.e. $\frac{-4+2i}{5}$

So, the simplified expression is  $\frac{-4+2i}{5}$.