Question

divide and simplify to the form a + bi.\\(\frac{52i}{5 + i}\\)

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $5-i$ (the complex conjugate of $5+i$) to eliminate the imaginary number in the denominator.
$$\frac{52i}{5+i} \times \frac{5-i}{5-i} = \frac{52i(5-i)}{(5+i)(5-i)}$$

Step2: Expand numerator

Use distributive property to expand the numerator. Recall that $i^2=-1$.
$$52i(5-i) = 260i - 52i^2 = 260i - 52(-1) = 52 + 260i$$

Step3: Expand denominator

Use difference of squares formula $(a+b)(a-b)=a^2-b^2$, and substitute $i^2=-1$.
$$(5+i)(5-i) = 5^2 - i^2 = 25 - (-1) = 26$$

Step4: Simplify the fraction

Divide each term in the numerator by the denominator.
$$\frac{52 + 260i}{26} = \frac{52}{26} + \frac{260i}{26}$$

Answer:

$2 + 10i$