Question

perform the indicated operations and write the result in standard form.
\\(\frac{7}{4 + \frac{7}{i}}\\)
\\(\frac{7}{4 + \frac{7}{i}} = \square\\)
(simplify your answer. type your answer in the form \\(a + bi\\). use integers or fractions for any numbers in the expression.)

Explanation:

Step1: Simplify the denominator first

First, simplify $\frac{7}{i}$. Multiply numerator and denominator by $i$ (since $i^2=-1$):
$\frac{7}{i} = \frac{7i}{i^2} = \frac{7i}{-1} = -7i$
Now the denominator becomes $4 + (-7i) = 4 - 7i$.
The original expression is now $\frac{7}{4 - 7i}$.

Step2: Rationalize the denominator

Multiply numerator and denominator by the conjugate of $4-7i$, which is $4+7i$:
$\frac{7(4+7i)}{(4-7i)(4+7i)}$

Step3: Expand numerator and denominator

Numerator:

$7(4+7i) = 28 + 49i$

Denominator (use $(a-b)(a+b)=a^2-b^2$):

$4^2 - (7i)^2 = 16 - 49i^2 = 16 - 49(-1) = 16 + 49 = 65$

Step4: Split into standard form

$\frac{28 + 49i}{65} = \frac{28}{65} + \frac{49}{65}i$

Answer:

$\frac{28}{65} + \frac{49}{65}i$