Let's solve the problem step by step.

We are given the expression:

$\frac{12 \sqrt{30}}{\sqrt{3}}$

Step 1: Simplify the surd division

Using the property of square roots: $\frac{\sqrt{a}}{\sqrt{b}} = \sqrt{\frac{a}{b}}$

$\frac{\sqrt{30}}{\sqrt{3}} = \sqrt{\frac{30}{3}} = \sqrt{10}.$

So the expression becomes: $12 \sqrt{10}.$

Final Answer:

The simplest form of the given expression is: $12 \sqrt{10}.$

Would you like additional explanations or have further questions? Here are 5 related questions to expand your understanding:

1. What are the basic rules of simplifying surds like $\sqrt{a} / \sqrt{b}$?
2. How can you rationalize a denominator if it contains a surd?
3. What are the properties of square roots used in algebraic manipulation?
4. What are the key differences between surds and integers in expressions?
5. Can you combine multiple surds, like $\sqrt{10} + 2\sqrt{10}$, and how?

Tip:

Always simplify square roots by factoring out perfect squares to make expressions easier to work with!