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The task is to find the modulus of the complex number \( z \), where:

\[
z = \frac{5i}{\sqrt{3} - i}
\]

### Steps to solve for \( |z| \):

1. **Rationalize the denominator**:
   To simplify the expression, multiply the numerator and the denominator by the conjugate of the denominator, \( \sqrt{3} + i \):

   \[
   z = \frac{5i}{\sqrt{3} - i} \times \frac{\sqrt{3} + i}{\sqrt{3} + i}
   \]

2. **Simplify the expression**:
   In the denominator, use the difference of squares:

   \[
   (\sqrt{3} - i)(\sqrt{3} + i) = \sqrt{3}^2 - i^2 = 3 - (-1) = 4
   \]

   Now the expression becomes:

   \[
   z = \frac{5i (\sqrt{3} + i)}{4}
   \]

   Expand the numerator:

   \[
   5i(\sqrt{3} + i) = 5i\sqrt{3} + 5i^2 = 5i\sqrt{3} - 5
   \]

   So:

   \[
   z = \frac{5i\sqrt{3} - 5}{4}
   \]

   This is the simplified form of \( z \):

   \[
   z = \frac{-5}{4} + \frac{5\sqrt{3}}{4}i
   \]

3. **Find the modulus**:
   The modulus of a complex number \( z = a + bi \) is \( |z| = \sqrt{a^2 + b^2} \).

   In this case:
   \[
   a = \frac{-5}{4}, \quad b = \frac{5\sqrt{3}}{4}
   \]

   So, the modulus is:

   \[
   |z| = \sqrt{\left( \frac{-5}{4} \right)^2 + \left( \frac{5\sqrt{3}}{4} \right)^2}
   \]

   Simplifying:

   \[
   |z| = \sqrt{\frac{25}{16} + \frac{75}{16}} = \sqrt{\frac{100}{16}} = \sqrt{6.25} = 2.5
   \]

Thus, the modulus of \( z \) is:

\[
|z| = 2.5
\]

---

Do you have any questions or would you like further details?

Here are some related questions for deeper understanding:
1. What is the significance of the modulus in complex numbers?
2. How does rationalizing the denominator affect the result?
3. What would the polar form of this complex number be?
4. How can we use this method for different complex number expressions?
5. Why is the difference of squares useful when simplifying complex numbers?

**Tip**: When multiplying complex numbers, always consider the conjugate to simplify expressions with imaginary units.

This guide walks you through finding the modulus of a complex number, z = (5i) / (sqrt(3) - i), by rationalizing the denominator and using the modulus formula. Perfect for Grades 10-12, with step-by-step explanations on rationalizing complex numbers and calculating modulus.

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