Question

question 5 of 10, step 1 of 1
simplify the following expression.
\\(\frac{2 + i}{5 + i}\\)
answer

Explanation:

Step1: Multiply by conjugate of denominator

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

Step2: Expand numerator using FOIL

Calculate product of numerator terms:
$$(2+i)(5-i) = 2\times5 + 2\times(-i) + i\times5 + i\times(-i) = 10 - 2i + 5i - i^2$$
Substitute $i^2=-1$:
$$10 + 3i - (-1) = 11 + 3i$$

Step3: Expand denominator (difference of squares)

Calculate product of denominator terms:
$$(5+i)(5-i) = 5^2 - i^2 = 25 - (-1) = 26$$

Step4: Combine numerator and denominator

Form simplified fraction:
$$\frac{11 + 3i}{26} = \frac{11}{26} + \frac{3}{26}i$$

Answer:

$\frac{11}{26} + \frac{3}{26}i$