Question

divide and simplify to the form a + bi.\\(\frac{104i}{5 + i}\\)\\(\frac{104i}{5 + i}=\square\\)(simplify your answer. use integers or fractions for any numbers in the expression. type your answer in a + bi.)

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $5-i$:
$$\frac{104i}{5+i} \times \frac{5-i}{5-i} = \frac{104i(5-i)}{(5+i)(5-i)}$$

Step2: Expand denominator (difference of squares)

Use $(a+b)(a-b)=a^2-b^2$, $i^2=-1$:
$$(5+i)(5-i) = 5^2 - i^2 = 25 - (-1) = 26$$

Step3: Expand numerator

Distribute $104i$ across $5-i$:
$$104i(5-i) = 104i \times 5 - 104i \times i = 520i - 104i^2$$
Substitute $i^2=-1$:
$$520i - 104(-1) = 104 + 520i$$

Step4: Divide by denominator

Divide each term by 26:
$$\frac{104 + 520i}{26} = \frac{104}{26} + \frac{520}{26}i$$

Step5: Simplify fractions

Reduce each fraction to lowest terms:
$$\frac{104}{26}=4, \quad \frac{520}{26}=20$$

Answer:

$4 + 20i$