Question

perform the indicated operations and write the result in standard form.\\(\frac{5}{3 + \frac{4}{i}}\\)\\(\frac{5}{3 + \frac{4}{i}} = \square\\)\\(\text{(simplify your answer. type your answer in the form } a + bi. \text{ use integers or fractions for any numbers in the expression.)}\\)

Explanation:

Step1: Simplify the denominator first

First, rewrite $\frac{4}{i}$ by multiplying numerator and denominator by $i$:
$\frac{4}{i} = \frac{4i}{i^2} = \frac{4i}{-1} = -4i$
Now the denominator becomes $3 - 4i$.
The expression is now $\frac{5}{3-4i}$.

Step2: Rationalize the denominator

Multiply numerator and denominator by the conjugate of $3-4i$, which is $3+4i$:
$\frac{5(3+4i)}{(3-4i)(3+4i)}$

Step3: Expand numerator and denominator

Numerator: $5\times3 + 5\times4i = 15 + 20i$
Denominator: Use $(a-b)(a+b)=a^2-b^2$: $3^2 - (4i)^2 = 9 - 16i^2 = 9 - 16(-1) = 9+16=25$
The expression becomes $\frac{15+20i}{25}$

Step4: Simplify the fraction

Split the fraction: $\frac{15}{25} + \frac{20}{25}i = \frac{3}{5} + \frac{4}{5}i$

Answer:

$\frac{3}{5} + \frac{4}{5}i$