https://file.aiquickdraw.com/mathbot/file/17294993004410rfl2vo2.jpg

The expression shown in the image is:

\[
\frac{2}{5 - 3i}
\]

To simplify this complex fraction, we need to eliminate the imaginary part in the denominator. This is done by multiplying both the numerator and the denominator by the **conjugate** of the denominator.

The conjugate of \(5 - 3i\) is \(5 + 3i\). So, let's multiply both the numerator and the denominator by \(5 + 3i\):

\[
\frac{2}{5 - 3i} \cdot \frac{5 + 3i}{5 + 3i} = \frac{2(5 + 3i)}{(5 - 3i)(5 + 3i)}
\]

Now, calculate the denominator using the difference of squares:

\[
(5 - 3i)(5 + 3i) = 5^2 - (3i)^2 = 25 - (-9) = 25 + 9 = 34
\]

So the denominator is 34.

Next, calculate the numerator:

\[
2(5 + 3i) = 2 \cdot 5 + 2 \cdot 3i = 10 + 6i
\]

Now we have:

\[
\frac{10 + 6i}{34}
\]

This can be simplified by dividing both the real and imaginary parts by 34:

\[
\frac{10}{34} + \frac{6i}{34} = \frac{5}{17} + \frac{3i}{17}
\]

Thus, the simplified form is:

\[
\frac{5}{17} + \frac{3i}{17}
\]

Or in rectangular form:

\[
\boxed{\frac{5}{17} + \frac{3i}{17}}
\]

Would you like further details on any part of this process or the theory behind it?

### Relative Questions:
1. What is the purpose of multiplying by the conjugate in complex numbers?
2. How do you derive the formula for the difference of squares involving imaginary numbers?
3. Can this fraction be represented in polar form?
4. How do you handle division of complex numbers in general?
5. What are some real-world applications of complex numbers?

### Tip:
Always check your final result by considering the original problem and confirming the calculation of both the real and imaginary components separately.

Learn how to simplify the complex expression 2 / (5 - 3i) by multiplying by the conjugate and using the difference of squares. A step-by-step solution with key concepts in complex numbers, suitable for Grades 10-12.

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