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{2 + 7i}{5 + 5i}=square)

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $5-5i$:
$$\frac{2+7i}{5+5i} \times \frac{5-5i}{5-5i} = \frac{(2+7i)(5-5i)}{(5+5i)(5-5i)}$$

Step2: Expand numerator via FOIL

Calculate product of numerator terms:
$$(2)(5) + (2)(-5i) + (7i)(5) + (7i)(-5i) = 10 -10i +35i -35i^2$$
Simplify using $i^2=-1$:
$$10 +25i -35(-1) = 10+25i+35 = 45+25i$$

Step3: Expand denominator (difference of squares)

Calculate product of denominator terms:
$$5^2 - (5i)^2 = 25 -25i^2$$
Simplify using $i^2=-1$:
$$25 -25(-1) = 25+25 = 50$$

Step4: Simplify the fraction

Split into real and imaginary parts, reduce fractions:
$$\frac{45+25i}{50} = \frac{45}{50} + \frac{25}{50}i = \frac{9}{10} + \frac{1}{2}i$$

Answer:

$\frac{9}{10} + \frac{1}{2}i$