Question

divide the following complex numbers and express the result in standard form, ( a + bi ), where ( a ) and ( b ) are fractions in lowest terms. (\frac{8 + 7i}{5 + i}) (\frac{8 + 7i}{5 + i}=square)

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $5-i$ (the complex conjugate of $5+i$):
$$\frac{8+7i}{5+i} \times \frac{5-i}{5-i} = \frac{(8+7i)(5-i)}{(5+i)(5-i)}$$

Step2: Expand numerator using FOIL

Calculate the product of the numerator terms:
$$(8+7i)(5-i) = 8\times5 + 8\times(-i) + 7i\times5 + 7i\times(-i) = 40 - 8i + 35i -7i^2$$
Since $i^2=-1$, substitute and simplify:
$$40 + 27i -7(-1) = 40 + 27i +7 = 47 + 27i$$

Step3: Expand denominator (difference of squares)

Calculate the product of the denominator terms, using $(a+b)(a-b)=a^2-b^2$:
$$(5+i)(5-i) = 5^2 - i^2 = 25 - (-1) = 26$$

Step4: Split into standard form

Divide the real and imaginary parts by the denominator:
$$\frac{47 + 27i}{26} = \frac{47}{26} + \frac{27}{26}i$$

Answer:

$\frac{47}{26} + \frac{27}{26}i$