Alright, let's simplify the given complex fraction step-by-step:
The given fraction is:
[tex]\[ \frac{-9 + 4i}{7i} \][/tex]
### Step 1: Multiply by the Conjugate of the Denominator
To simplify this fraction, we need to eliminate the imaginary unit [tex]\(i\)[/tex] from the denominator. We do this by multiplying both the numerator and the denominator by the conjugate of the denominator. For [tex]\(7i\)[/tex], the conjugate is [tex]\(-7i\)[/tex].
Thus, we have:
[tex]\[ \frac{-9 + 4i}{7i} \cdot \frac{-7i}{-7i} \][/tex]
This multiplication will be carried out both in the numerator and the denominator.
### Step 2: Multiply the Numerator by the Conjugate of the Denominator
Calculate the new numerator:
[tex]\[ (-9 + 4i) \cdot (-7i) \][/tex]
Using the distributive property (FOIL method):
[tex]\[ (-9) \cdot (-7i) + (4i) \cdot (-7i) \][/tex]
[tex]\[ = 63i + (-28i^2) \][/tex]
We know that [tex]\(i^2 = -1\)[/tex], so substituting [tex]\(i^2\)[/tex]:
[tex]\[ 63i - 28(-1) \][/tex]
[tex]\[ = 63i + 28 \][/tex]
Thus, the new numerator is:
[tex]\[ 28 + 63i \][/tex]
### Step 3: Multiply the Denominator by its Conjugate
Calculate the new denominator:
[tex]\[ (7i) \cdot (-7i) \][/tex]
[tex]\[ = -49i^2 \][/tex]
Again, substituting [tex]\(i^2 = -1\)[/tex]:
[tex]\[ = -49(-1) \][/tex]
[tex]\[ = 49 \][/tex]
The new denominator is:
[tex]\[ 49 - 0i \][/tex]
### Step 4: Simplify the Fraction
Now we have:
[tex]\[ \frac{28 + 63i}{49} \][/tex]
We can separate the real and imaginary parts:
[tex]\[ = \frac{28}{49} + \frac{63i}{49} \][/tex]
Simplify both parts:
[tex]\[ = \frac{28}{49} + \frac{63}{49}i \][/tex]
[tex]\[ = \frac{4}{7} + \frac{9}{7}i \][/tex]
### Step 5: Write the Final Answer
Combine the simplified parts:
[tex]\[ = 0.5714285714285714 + 1.2857142857142858i \][/tex]
So, the simplified form of the given complex fraction [tex]\(\frac{-9+4i}{7i}\)[/tex] is:
[tex]\[ 0.5714285714285714 + 1.2857142857142858i \][/tex]
The final result, after all these steps, gives us:
[tex]\[ \left( 28 + 63i, 49, 0.5714285714285714 + 1.2857142857142858i \right) \][/tex]
