Question

watch the video and then solve the problem given below
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divide and simplify to the form a + bi.
\\(\frac{60i}{3 + i}\\)
\\(\frac{60i}{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 by conjugate of denominator

Multiply numerator and denominator by $3-i$ (the conjugate of $3+i$):
$$\frac{60i}{3+i} \times \frac{3-i}{3-i} = \frac{60i(3-i)}{(3+i)(3-i)}$$

Step2: Expand numerator and denominator

Expand using distributive property and $i^2=-1$:
Numerator: $60i \times 3 - 60i \times i = 180i - 60i^2 = 180i - 60(-1) = 60 + 180i$
Denominator: $3^2 - i^2 = 9 - (-1) = 10$
Expression becomes: $\frac{60 + 180i}{10}$

Step3: Split and simplify fractions

Divide each term in numerator by denominator:
$$\frac{60}{10} + \frac{180i}{10}$$

Answer:

$6 + 18i$