Question

divide the complex number. write the answer in the form a + bi. express numbers in exact form. \\(\frac{4 + 4i}{7 + 4i} = \square\\)

Explanation:

Step1: Multiply numerator and denominator by the conjugate of the denominator.

The conjugate of \(7 + 4i\) is \(7 - 4i\). So we multiply \(\frac{4 + 4i}{7 + 4i}\) by \(\frac{7 - 4i}{7 - 4i}\).

Step2: Expand the numerator and the denominator.

Numerator: \((4 + 4i)(7 - 4i)=4\times7 - 4\times4i + 4i\times7 - 4i\times4i = 28 - 16i + 28i - 16i^{2}\). Since \(i^{2}=-1\), this becomes \(28 + 12i + 16 = 44 + 12i\).
Denominator: \((7 + 4i)(7 - 4i)=7^{2}-(4i)^{2}=49 - 16i^{2}=49 + 16 = 65\).

Step3: Write the result in \(a + bi\) form.

\(\frac{44 + 12i}{65}=\frac{44}{65}+\frac{12}{65}i\).

Answer:

\(\frac{44}{65}+\frac{12}{65}i\)