Let's simplify the expression $2\sqrt{12} - 3\sqrt{45}$.

Step 1: Simplify $\sqrt{12}$

We can factor $12$ as $4 \times 3$, and since $\sqrt{4} = 2$, we get: $\sqrt{12} = \sqrt{4 \times 3} = \sqrt{4} \times \sqrt{3} = 2\sqrt{3}.$ So, $2\sqrt{12} = 2 \times 2\sqrt{3} = 4\sqrt{3}.$

Step 2: Simplify $\sqrt{45}$

We can factor $45$ as $9 \times 5$, and since $\sqrt{9} = 3$, we get: $\sqrt{45} = \sqrt{9 \times 5} = \sqrt{9} \times \sqrt{5} = 3\sqrt{5}.$ So, $3\sqrt{45} = 3 \times 3\sqrt{5} = 9\sqrt{5}.$

Step 3: Combine the terms

Now we substitute the simplified values into the original expression: $2\sqrt{12} - 3\sqrt{45} = 4\sqrt{3} - 9\sqrt{5}.$ Since the terms involve different square roots ($\sqrt{3}$ and $\sqrt{5}$), they cannot be simplified further. So, the simplified expression is: $4\sqrt{3} - 9\sqrt{5}.$

Final Answer:

$4\sqrt{3} - 9\sqrt{5}.$

Would you like more details or have any questions about this? Here are 5 related questions you might find useful:

1. How do you simplify square roots in expressions like $\sqrt{72}$ or $\sqrt{200}$?
2. What is the general method for simplifying radical expressions?
3. How can you add or subtract square roots with different radicands?
4. What happens when you encounter a negative sign in front of square roots?
5. Can square roots be simplified using prime factorization?

Tip: Remember that simplifying square roots often involves breaking down the number under the radical into its prime factors.