Let's simplify the expression:

$\sqrt{6} - 2\sqrt{24} + 3\sqrt{96}$

Step 1: Simplify each square root

• $\sqrt{6}$ stays as it is (it cannot be simplified further).
• $\sqrt{24} = \sqrt{4 \times 6} = \sqrt{4} \times \sqrt{6} = 2\sqrt{6}$.
• $\sqrt{96} = \sqrt{16 \times 6} = \sqrt{16} \times \sqrt{6} = 4\sqrt{6}$.

Step 2: Substitute the simplified square roots back into the expression:

$\sqrt{6} - 2(2\sqrt{6}) + 3(4\sqrt{6})$

Step 3: Multiply and combine like terms:

• $2(2\sqrt{6}) = 4\sqrt{6}$,
• $3(4\sqrt{6}) = 12\sqrt{6}$.

Now, substitute these into the expression: $\sqrt{6} - 4\sqrt{6} + 12\sqrt{6}$

Step 4: Combine like terms:

$(\sqrt{6} - 4\sqrt{6} + 12\sqrt{6}) = (1\sqrt{6} - 4\sqrt{6} + 12\sqrt{6}) = 9\sqrt{6}$

Final answer:

$9\sqrt{6}$

Would you like further details or have any questions?

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Here are 5 related questions to explore further:

1. How do you simplify square roots with different numbers inside?
2. Can you explain how to combine like terms with square roots?
3. What is the difference between simplifying a square root and factoring it?
4. How do square roots behave with multiplication and division?
5. How can we simplify more complex radical expressions?

Tip: Always check if the number inside a square root can be factored into a perfect square to simplify the expression.