Question

question 3 of 10, step 1 of 1
simplify the following expression.
\\(\dfrac{41}{4 + 5i}\\)
answer

Explanation:

Step1: Multiply by conjugate of denominator

Multiply numerator and denominator by $4-5i$:
$\frac{41}{4+5i} \times \frac{4-5i}{4-5i} = \frac{41(4-5i)}{(4+5i)(4-5i)}$

Step2: Expand denominator using difference of squares

Use $(a+b)(a-b)=a^2-b^2$, $i^2=-1$:
$(4+5i)(4-5i)=4^2-(5i)^2=16-25i^2=16+25=41$

Step3: Expand numerator

Calculate $41(4-5i)$:
$41\times4 - 41\times5i = 164 - 205i$

Step4: Divide numerator by denominator

Divide by the simplified denominator:
$\frac{164 - 205i}{41} = \frac{164}{41} - \frac{205i}{41}$

Step5: Simplify the fractions

Reduce each term to lowest terms:
$4 - 5i$

Answer:

$4 - 5i$