Question

divide and simplify to the form a + bi.
\\(\frac{20i}{3 + i}\\)
\\(\frac{20i}{3 + i} = \square\\)
(simplify your answer. use integers or fractions for any numbers in the expression. type your answer in the form a + bi.)

Explanation:

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

The conjugate of \(3 + i\) is \(3 - i\). So we multiply \(\frac{20i}{3 + i}\) by \(\frac{3 - i}{3 - i}\):

$$ \frac{20i(3 - i)}{(3 + i)(3 - i)} $$

Step2: Expand the numerator and the denominator.

First, expand the numerator: \(20i(3 - i)=60i - 20i^{2}\). Since \(i^{2}=- 1\), this becomes \(60i-20(-1)=20 + 60i\).
Then, expand the denominator using the difference of squares formula \((a + b)(a - b)=a^{2}-b^{2}\), where \(a = 3\) and \(b = i\): \((3 + i)(3 - i)=3^{2}-i^{2}=9-(-1)=10\).

Step3: Simplify the fraction.

Now we have \(\frac{20 + 60i}{10}\). Divide each term in the numerator by 10: \(\frac{20}{10}+\frac{60i}{10}=2 + 6i\).

Answer:

\(2 + 6i\)