Question

simplify the following expression.
\\(\frac{8 + 7i}{8 - 7i}\\)
answer

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $8+7i$:
$$\frac{8+7i}{8-7i} \times \frac{8+7i}{8+7i} = \frac{(8+7i)^2}{(8-7i)(8+7i)}$$

Step2: Expand numerator

Use $(a+b)^2=a^2+2ab+b^2$ and $i^2=-1$:
$$(8+7i)^2 = 8^2 + 2\times8\times7i + (7i)^2 = 64 + 112i + 49i^2 = 64 + 112i - 49 = 15 + 112i$$

Step3: Expand denominator

Use difference of squares $(a-b)(a+b)=a^2-b^2$:
$$(8-7i)(8+7i) = 8^2 - (7i)^2 = 64 - 49i^2 = 64 + 49 = 113$$

Step4: Combine results

Divide the simplified numerator by denominator:
$$\frac{15 + 112i}{113}$$

Answer:

$\frac{15}{113} + \frac{112}{113}i$