Express `(13i)/(2-3i)` in the form `A + Bi`

Answer

To express \(\frac{13i}{2 - 3i}\) in the form \(A + Bi\), we will follow these steps:

### Step 1: Multiply by the Conjugate
To simplify the expression, we multiply the numerator and the denominator by the conjugate of the denominator. The conjugate of \(2 - 3i\) is \(2 + 3i\).

\[ \frac{13i}{2 - 3i} \cdot \frac{2 + 3i}{2 + 3i} = \frac{13i(2 + 3i)}{(2 - 3i)(2 + 3i)} \]

### Step 2: Simplify the Denominator
Now, we simplify the denominator using the difference of squares formula:

\[ (2 - 3i)(2 + 3i) = 2^2 - (3i)^2 = 4 - 9(-1) = 4 + 9 = 13 \]

### Step 3: Expand the Numerator
Next, we expand the numerator:

\[ 13i(2 + 3i) = 13i \cdot 2 + 13i \cdot 3i = 26i + 39i^2 \]

Since \(i^2 = -1\), we can substitute:

\[ 39i^2 = 39(-1) = -39 \]

Thus, the numerator becomes:

\[ 26i - 39 \]

### Step 4: Combine the Results
Now we can combine the results:

\[ \frac{26i - 39}{13} \]

### Step 5: Separate Real and Imaginary Parts
We can separate the real and imaginary parts:

\[ \frac{-39}{13} + \frac{26i}{13} = -3 + 2i \]

### Final Result
Thus, we express \(\frac{13i}{2 - 3i}\) in the form \(A + Bi\):

\[ A + Bi = -3 + 2i \]

Where \(A = -3\) and \(B = 2\).

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