what is the quotient of the complex number 4 - 3i divided by its conjugate?

Explanation:

Step1: Find the conjugate

The conjugate of the complex number (4 - 3i) is (4+3i).

Step2: Set up the division

We need to find (\frac{4 - 3i}{4 + 3i}). Multiply the numerator and denominator by the conjugate of the denominator (4 - 3i). So we have (\frac{(4 - 3i)(4 - 3i)}{(4 + 3i)(4 - 3i)}).

Step3: Expand the numerator and denominator

Expand the numerator ((4 - 3i)(4 - 3i)=16-12i-12i + 9i^{2}=16-24i-9=7-24i) (since (i^{2}=-1)). Expand the denominator ((4 + 3i)(4 - 3i)=16-9i^{2}=16 + 9=25) (using the difference - of - squares formula ((a + b)(a - b)=a^{2}-b^{2}) with (a = 4) and (b = 3i)).

Step4: Write the result

The quotient is (\frac{7-24i}{25}=\frac{7}{25}-\frac{24}{25}i).

Answer:

(\frac{7}{25}-\frac{24}{25}i) (the third option)