Question

use the complex conjugate to divide these complex numbers.\\(\frac{5 + 4i}{6 - 2i}\\)\\(\frac{?}{\square}+\frac{\square}{\square}i\\)remember to simplify.

Explanation:

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

The complex conjugate of \(6 - 2i\) is \(6 + 2i\). So we multiply the fraction \(\frac{5 + 4i}{6 - 2i}\) by \(\frac{6 + 2i}{6 + 2i}\):

$$ \frac{(5 + 4i)(6 + 2i)}{(6 - 2i)(6 + 2i)} $$

Step2: Expand the numerator and the denominator.

First, expand the numerator using the distributive property (FOIL method):

$$
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$$

Since \(i^2=-1\), we substitute that in:

$$ 30 + 10i + 24i + 8\times(-1)=30 + 34i - 8 = 22 + 34i $$

Now, expand the denominator using the difference of squares formula \((a - b)(a + b)=a^2 - b^2\), where \(a = 6\) and \(b = 2i\):

$$
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$$

Substitute \(i^2 = - 1\):

$$ 36 - 4\times(-1)=36 + 4 = 40 $$

Step3: Write the fraction as a complex number in standard form.

Now our fraction is \(\frac{22 + 34i}{40}\). We can split this into the real and imaginary parts:

$$ \frac{22}{40}+\frac{34}{40}i $$

Simplify the fractions:

$$ \frac{11}{20}+\frac{17}{20}i $$

Answer:

\(\frac{11}{20}+\frac{17}{20}i\) (So the real part numerator is \(11\), real part denominator is \(20\), imaginary part numerator is \(17\), imaginary part denominator is \(20\))