Let's simplify the expression:

\[
\frac{13i}{10i(5 - 3i)}
\]

### Step 1: Simplify the denominator
Multiply the denominator:

\[
10i(5 - 3i) = 10i \cdot 5 - 10i \cdot 3i = 50i - 30i^2
\]

Recall that \(i^2 = -1\), so:

\[
50i - 30(-1) = 50i + 30 = 30 + 50i
\]

Thus, the expression becomes:

\[
\frac{13i}{30 + 50i}
\]

### Step 2: Rationalize the denominator
To remove the imaginary part from the denominator, multiply both the numerator and the denominator by the conjugate of the denominator, \(30 - 50i\):

\[
\frac{13i}{30 + 50i} \times \frac{30 - 50i}{30 - 50i} = \frac{13i(30 - 50i)}{(30 + 50i)(30 - 50i)}
\]

### Step 3: Multiply the numerator and simplify
First, let's expand the numerator:

\[
13i(30 - 50i) = 13i \cdot 30 - 13i \cdot 50i = 390i - 650i^2
\]

Since \(i^2 = -1\):

\[
390i - 650(-1) = 390i + 650 = 650 + 390i
\]

### Step 4: Multiply the denominator
Now expand the denominator:

\[
(30 + 50i)(30 - 50i) = 30^2 - (50i)^2 = 900 - 2500i^2
\]

Since \(i^2 = -1\):

\[
900 - 2500(-1) = 900 + 2500 = 3400
\]

### Step 5: Final answer
Now, the expression becomes:

\[
\frac{650 + 390i}{3400}
\]

We can simplify this by dividing both the real and imaginary parts by 3400:

\[
\frac{650}{3400} + \frac{390i}{3400} = \frac{65}{340} + \frac{39i}{340} = \frac{13}{68} + \frac{39i}{340}
\]

Thus, the simplified form of the expression is:

\[
\frac{13}{68} + \frac{39i}{340}
\]

### Do you want more details or have any questions?

Here are 5 related questions to practice:
1. Simplify \(\frac{5i}{2i(4 - i)}\).
2. Rationalize and simplify \(\frac{3 + 2i}{1 - i}\).
3. Find the conjugate of \(\frac{4 + i}{3 - 2i}\).
4. Simplify \(\frac{2 + 3i}{1 + 2i}\).
5. Multiply \((2 + i)(3 - 4i)\) and simplify.

**Tip:** Always remember that \(i^2 = -1\) when simplifying complex numbers.

This solution demonstrates how to simplify the complex number expression 13i/10i(5-3i) by rationalizing the denominator and multiplying by the conjugate. Learn key steps in handling complex number multiplication and division, with a detailed breakdown suitable for high school students (Grades 10-12).

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